Topics in Nonlinear Partial Differential Equations

Csirik Mihaly

Abstract

Contents1 Motivation – Classical Theory 2

2 Measure Theory 52.1 Lebesgue’s Integration Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Lp spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Maximal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Modes of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Lattice Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Caratheodory’s Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 Lebesgue and Hausdorff Measures . . . . . . . . . . . . . . . . . . . . . . . . 142.8 Borel and Radon Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.9 Young Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Analysis in Vector Spaces 153.1 Hyperplanes. The Hahn–Banach Theorem and its Corollaries . . . . . . . . . . 153.2 Topological Dual Space of a Normed Space . . . . . . . . . . . . . . . . . . . 173.3 Reflexive spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Uniformly Convex Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Strictly Convex Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 Subdifferentials. Duality Mappings . . . . . . . . . . . . . . . . . . . . . . . . 223.7 Eigenstructure of Compact Operators . . . . . . . . . . . . . . . . . . . . . . 233.8 Courant–Fischer Minimax Principles . . . . . . . . . . . . . . . . . . . . . . . 27

4 Weak topologies 324.1 Locally Convex Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Separation of Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Weak and Weak-* Topologies as Product Topologies . . . . . . . . . . . . . . 354.4 Weak and Weak-* Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 374.5 Weak and Weak-* Compactness . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Distributions 425.1 Topology on Spaces of Continuous Functions . . . . . . . . . . . . . . . . . . 425.2 The Riesz–Kakutani Representation Theorem . . . . . . . . . . . . . . . . . . 45

1

6 Sobolev spaces 466.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.2 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.3 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.4 Riesz Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7 Linear Elliptic Boundary Value Problems 527.1 The Courant–Fischer Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8 Critical Point Theory 528.1 Variational Principles and Compactness . . . . . . . . . . . . . . . . . . . . . 528.2 ??? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

9 Nonlinear Problems 56

10 Semicontinuity and Quasiconvexity 5610.1 Nemytskii Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

11 Numerical Methods 6011.1 The Newton–Kantorovich Theory . . . . . . . . . . . . . . . . . . . . . . . . 6011.2 Multilevel Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

1 Motivation – Classical Theory

1. Calculus of varations is classical field of analysis pioneered by Bernoulli, Euler and La-grange. In the late 19th century however, Weierstrass demonstrated that a minimization prob-lem does not necessarily have a solution. This led to the abandonment of the use of Dirichlet’sprinciple in complex analysis, and a suspicious attitude towards variational methods.

In the beginning of the 20th century Hilbert and Tonelli finally gave the field a clear footing.They recognized the importance of convexity of the energy functional, in particular Dirichlet’sintegral. In this section we present these semi-classical results as a motivation for digging intothe theory of reflexive Banach spaces, weak topology and Sobolev spaces. Luckily for us, thisthen-heuristic field now has a beautiful theory, both simple and immensly powerful for studyingpartial differential equations and related topics.

The following classical theorem is taken form Dracorogna, and presented with a proof withforward references to the definitions.(1.1) Theorem. Let � � Rn be a Lipschitz domain, and L 2 C 1.� � R � Rn/, such that

(1) .�; �/ 7! L.x; �; �/ is convex for every x 2 �.

(2) There exists p > 0, c1 > 0, and c2, such that

8x 2 � 8� 2 R 8� 2 Rn W jL.x; �; �/j � c1k�kpC c2:

(3) There exists c3 � 0, such that with the p in (2), we have

8x 2 � 8� 2 R 8� 2 Rn Wˇ@L.x; �; �/

@�

ˇ; @L.x; �; �/

@�

� c3�1C j�jp�1 C k�kp�1�:2

Statement. The minimization problem

j.u/ D min˚j.v/ W v 2 g CW

1;p0 .�/

;

where g 2 W 1;p.�/, j.g/ < C1 and

j.v/ D

Z�

L.x; v;Dv/ dx

has a solution u 2 g C W 1;p0 .�/. Moreover, if (1) is improved to strict convexity than the

solution is unique.

(1.2) Lemma. The energy functional j is weakly sequentially lower semicontinuous, i.e.

8fu˛g � W1;p.�/ 8u 2 W 1;p.�/ W u˛ * u in W 1;p.�/ H) lim inf j.u˛/ � j.u/

Proof of the lemma. Since L 2 C 1.� � R � Rn/, and .�; �/ 7! L.x; �; �/ is convex, we havea supporting hyperplane inequality at the point .u Du/ 2 R � Rn for a weakly convergentsequence u˛ * u in W 1;p.�/

L.x; u˛;Du˛/ � L.x; u;Du/ �@L.x; u;Du/

@�.u˛ � u/C

@L.x; u;Du/

@�;Du˛ �Du

!:

Formally integrate the inequality,

j.u˛/ � j.u/ �

Z�

@L.x; u;Du/

@�.u˛ � u/ dx C

Z�

@L.x; u;Du/

@�;Du˛ �Du

!dx

therefore we should prove that the right hand side tends to 0 as ˛ !1. This follows from theweak convergence u˛ * u in W 1;p.�/, since that is equivalent to statment u˛ * u in Lp.�/and Du˛ *Du in ŒLp.�/�d . Thus we have to show that

@L. � ; u;Du/

@�2 Lp

0

.�/;@L. � ; u;Du/

@�2 ŒLp

0

.�/�d ;

thereby justifying the integration via Holder’s inequality, and also the allusion to weak conver-gence, since .Lp/ Š Lp

0

.Let us p0-integrate the growth condition (3) to getZ�

ˇ@L.x; u;Du/@�

ˇp0dx � c

p0

3

Z�

�1C jujp�1 C kDukp�1

�p=.p�1/dx

� C

Z�

1C jujp C kDukp dx D C�j�j C kuk

p

W 1;p

�< C1:

To establish the last inequality simply note that the .p�1/-norm and the p-norm are equivalentin R3, and consider the vector .1; juj; kDuk/.Proof of existence. The plan is to show that an energy-minimizing sequence has a weakly con-vergent subsequence, then via the weak lower semicontinuity of j , we obtain that the aformen-tioned weak limit is in fact a minimizer.

First, note that due to (2), a minimizer has a finite energy

C1 > j.g/ � K > �1;

3

whereK D inf

˚j.v/ W v 2 g CW

1;p0 .�/

:

Let fu˛g � g CW1;p0 .�/ be a minimizing sequence, that is

j.u˛/! K:

From the growth condition (2) we have after integration

K C 1 � j.u˛/ � c1kDu˛kpp C c2j�j;

where the upper estimate holds for sufficently large ˛ 2 N. Therefore fDu˛g is bounded inLp.�/. Applying Poincare’s inequality to the function .u˛ � g/, we get

C � kDu˛kp C kDgkp � kDu˛ �Dgkp � c�ku˛ � gkW 1;p � c��ku˛kW 1;p � kgkW 1;p

�:

Thus fu˛g is bounded in W 1;p.�/, the weak sequential compactness of W 1;p.�/ implies thatthere exists a subseqence fuˇ g � fu˛g and u 2 W 1;p.�/, such that uˇ * u in W 1;p.�/.

From the weak sequential lower semicontinuity of j , we get

lim inf j.uˇ / � j.u/;

so K � j.u/. But from the definition of K, we have K � j.u/, so j.u/ D K, proving theexistence of a minimizer.

It is highly important to note that the above compactness argument couldn’t possibly workjust by working with strong (or norm) convergence. This is because in infinite dimensionsa closed ball cannot be compact. This is a serious difficulty, that requires the coarsening ofstrong topology. But this leads nonmetrizable spaces, thereby requiring the use of general, set-theoretic topology. Fortunately, sequences will suffice to describe the weak topology, but thisis highly nontrivial.

Weak topologies proved to be powerful instruments in analysing Banach spaces, so we havea very satisfying bird’s eye (i.e. functional analysis) view of the whole situation.

As another example showing the necessity of coarsening the norm topology, let us lookat Weierstrass’ theorem or the extreme value theorem. It constitutes one of the most efficientinstruments of elementary analysis.

(1.3) Theorem. Let .X; d/ be a compact topological space, f W X �! R continuous. Then

9a; b 2 X 8x 2 X W infXf D f .a/ � f .x/ � f .b/ D sup

X

f:

From the point of view of general topology, it is trivial: the continuous image of a compactspace is compact. In finite dimensions, the Heine–Borel theorem informs us about when thetheorem is applicable: compact sets in Rd are precisely those which are bounded and closed.

But compact sets are ”thin” in infinite dimensions as one of Riesz’ early observations tellus. It also sheds light on one of the main reasons why the complications that aries in infinitedimensions, in fact this lemma can be though of as an essential building block of functionalanalysis.

(1.4) Riesz’s lemma. Let X be a normed space. Then B.0; 1/ is compact if and only if X isfinite dimensional.

Why we said that compact sets are ”thin”? Because if K � X is compact, then intK D ;,since a finite dimensional subspace has empty interior.

4

2 Measure TheoryIn this section we present the absolute essentials of measure theory. We shall do so withoutappealing to any functional analysis (c.f. Rudin), nor going to the absolute extremes of abstrac-tion (cf. Bourbaki). Since our underlying space will be Rn, or a bounded open subset of it, weshall assume that

X is a locally compact, separable metric space.

Nevertheless remind the reader of this assumption from time to time.

Figure 1: Roadmap of convergences.

2.1 Lebesgue’s Integration TheoryMeasurable functions. Let .X;F ; �/ be a measure space, where dom� D F a � -algebra and� is a measure. A function f W X �! R [ fC1g is called measurable if for every Borel setB 2 B.R/, the preimage is �-measurable: f �1.B/ 2 F . Here, the symbol B.R/ denotes thesmallest � -algebra generated by the open subsets of R. It is customary to explicitly write ”letf be measurable”; however, the reader should know that the construction of a non-measurablefunction requires the axiom of choice.

Measurability of f is equivalent to the measurability of the level sets ff � ag, or ff � ag,etc. for every a 2 R. It is also equivalent to the measurability of the preimages f �1.U /, whereU � R is open. Therefore composition of a measurable function with a continuous R �! Ris still measurable. Characteristic functions of measurable sets, finite linear combinations,products, inf’s and sup’s, lim sup’s and lim inf’s of measurable functions are again measurable– as long as the families under consideration are countable. It follows that nonnegative sums,positive/negative parts, absolute values are measureable, as well as limits – if they exist, and ifthe sets where they exist are measurable.

5

Step functions. For every measurable function f W X �! R there exists a sequenceof step functions ff˛g, (i.e., each can be written as a finite linear combination of characteristicfunctions of (disjoint) �-measurable sets) such that f˛ converges pointwise to f (onX ). More-over it is possible to choose the sequence ff˛g, in a way that jf˛j % jf j (on X ) is achieved.This is called pointwise approximation by step functions.

Integral. Let J be a finite index set, fcˇ g � RC and fEˇ g � F , so that

f DXˇ2J

cˇ�Eˇ

is a step function. Then the integral of f W X �! RC is defined asZf d� D

Xˇ2J

cˇ�.E˛/;

an additive, positive, positive homoegeneous functional on the vector lattice of nonnegativestep functions. The integral is insensitive to the particular representation of the function f , andas mentioned before, fEˇ g can be made disjoint. Using positive-negative decomposition, it ispossible to extend this functional to arbitrary step functions f , if

RfC d� and

Rf� d� are not

simultaneouslyC1, then Zf d� D

ZfC d� �

Zf� d�:

Given a nonnegative measurable function f , construct a sequence ffˇ g of step functions con-verging pointwise to f and defineZ

f d� D supn Z

fˇ d� W ˇ 2 No:

Of course such a concept would be utterly useless if the integral’s value depends on the partic-ular choice of the approximating sequence. Fortunately it is well-defined as-is and admits anextenstion to arbitrary measurable functions.

A measurable function f W X �! R is called �-integrable ifRfC d� and

Rf� d� are not

simultaneously C1. Note that every measurable function is integrable. Moreover, f called�-summable, if

Rjf j d� < C1, the class of such functions is denoted by L1.X;F ; �/ or

simply L1.X/. A useful generalization of this class is the set L1loc.X/ of locally summable

funcions, i.e. for whichRKjf j d� < C1 for every compact subset K � X . This last concept

of course necessitates a topology on X .Beppo Levi’s theorem, or the monotone convergence theorem is a typical first building

block for the further development of the integral. It is equivalently good to begin with Fatou’slemma.

(2.1) Beppo Levi’s theorem. Let ff˛g be a monotone sequence of nonnegative, measurablefunctions, pointwise converging to a function f . Then f is measurable (we already knew this),and Z

f˛ d�%

Zf d�:

In other words, the two sup’s – one in the defintion in the integral, and the other in takingthe monotone limit – are interchangeable. This fact is the mother of all integral convegencetheorems.

6

(2.2) Fatou’s lemma.

(1) Let ff˛g be a sequence of nonnegative, measurable functions. ThenZlim inf˛2N

f˛ d� � lim inf˛2N

Zf˛ d�:

(2) Let ff˛g be a sequence of measurable functions, and g an integrable functiondominating ff˛g, i.e. f˛ � g for all ˛ 2 N. Then

lim sup˛2N

Zf˛ d� �

Zlim sup˛2N

f˛ d�:

(2.3) Refined Fatou’s lemma. Let ff˛g � Lp.X;F ; �/, f˛ ! f a.e., where 0 < p < 1.Then Z ˇ

jf˛jp� jf jp � jf˛ � f j

pˇd�! 0 .˛ !1/:

(2.4) Corollary. Let ff˛g � Lp.X;F ; �/, where 0 < p <1. Then

kf˛kp ! kf kp; and f˛ ! f a.e. H) f 2 Lp; and kf˛ � f kp ! 0:

Proof. Rewrite the refined Fatou lemma using the reverse triangle inequality asZ ˇjf˛j

p� jf jp

ˇd� �

Zjf˛ � f j

p d�! 0;

from which the result follows.(2.5) Remark. It is worth recording the importance of this result in a somewhat more intuitiveform:

kf˛kp D kf kp C kf˛ � f kp C o.1/;

where o.1/ is denotes the set of functions tending to 0, as ˛ ! 1. In words, this meansthat in the limit, the norms fkf˛kpg of the individual elements of an Lp-sequnce decoupleinto the norm of the limit kf kp and the distance kf˛ � f kp. This explains how the functionsf˛ D �.˛;˛C1/ fail to converge in Lp.R/. Of course f˛ ! 0 a.e., therefore the only candidateis f � 0 a.e., however the quantity kf˛kp is not really ”flowing” form kf˛ � f kp to kf kp.There are neat applications of this refined version of Fatou’s lemma, discovered by Brezis andLieb.

Next, the dominated convergence theorem, or Lebesgue’s theorem refines part (2) of Fatou’slemma.

(2.6) Lebesgue’s theorem. Let ff˛g be a sequence of measurable functions, such that f˛ ! f

a.e. Suppose that there exists a summable function g dominating ff˛g, that is, a summablefunction g, such that jf˛j � g holds for every ˛ 2 N. ThenZ

jf˛ � f j d�! 0:

7

(2.1) Example. Let 1 � p < 1. Consider a countable dense subset fx˛g of Œ0; 1� (take, forexample the set of dyadic fractions), and let

f˛.x/ DX1�ˇ�˛

2�ˇ log jx � xˇ j:

Then each f˛ are (improper) Riemann integrable, sinceZ xˇ�"

0

log.xˇ � x/ dx D �" log "C "C xˇ .log xˇ � 1/! xˇ .log xˇ � 1/Z 1

xˇC"

log.x � xˇ / dx D .1 � xˇ /.log.1 � xˇ / � 1/ � " log "C "! .1 � xˇ /.log.1 � xˇ / � 1/

thus Z 1

0

log.x � xˇ / dx D xˇ .log xˇ � 1/C .1 � xˇ /.log.1 � xˇ / � 1/:

But the pointwise limit function

f .x/ DXˇ�1

2�ˇ log jx � xˇ j

is not (improper) Riemann integrable for if one takes an arbitrary subdivision of .0; 1/, nomatter how fine it is, every subinterval will contain a point x , where f blows up to �1.Therefore the lower integrals are all �1 and the upper integrals are converging to 0.

Lebesgue’s theory offers a remedy to this situation. First, let us prove convergence in theLp norm,

kf˛ � f kp D Xˇ�˛

2�ˇ log j � �xˇ j p�

Xˇ�˛

2�ˇ log j � �xˇ j

p

� 2�˛C1 supˇ�1

log j � �xˇ j p! 0:

since log j � �xˇ j 2 Lp.0; 1/. This last fact follows using interpolation of trivial cases p 2 N,see Lemma (2.11). Note that f˛ is a linear combination of functions in Lp.0; 1/, hence f˛ 2Lp.0; 1/. But Lp.0; 1/ is complete, so f 2 Lp.0; 1/ too.

(2.7) Bathtub principle. Let X be a measure space with measure �, and suppose that f WX �! R is a measurable function such that �.ff < tg/ <1 for all t 2 R. For given G > 0,define the set of functions

C Dng W X �! R W 0 � g � 1;

Zg d� D G

o:

Statement. The problem of finding g 2 C , such thatZgf d� D inf

h2C

Zhf d�

is solved by g D �ff <sg C c�ffDsg, where

s D supft 2 R W �.ff < tg/ � Gg;

�.ff < sg/C c�.ff D sg/ D G:

8

(2.8) Remark. Some explanation as to why is the above called ”bathtub principle”. Imagineas we start to fill up the set X � R with water from the ”bottom”. At a sufficiently high waterlevel t 2 R, the water level rises to the graph of the function f and suppose that it is made ofsome porous material, so that water can freely penetrate it. Then the amount of water above thegraph of f is

ft D

Zff�tg

.t � f / d�:

The bathtub principle tells us that if we only let a finite volume of waterG to penetrate the graphof f , what the depth of the water will be. A function h 2 C is ought to describe this: givena point x 2 X , instead of the natural water depth t � f .x/, the depth will be h.x/.t � f .x//.Physical intution tells us that h should be is either one or zero. Nevertheless, the amount ofwater described by such a distribution h is

ht D

Zfhf�htg

.ht � hf / d�:

But h was required to have integral G and 0 � h � 1, so

ht D Gt �

Zff�tg

hf d�:

The question is that given that the water level is

s D supft 2 R W �.ff < tg/ � Gg;

and the volume we are letting inside is G, what function h will render hs minimal? We claimthat this is precisely the question answered by the bathtub principle, because

suph2C

hs D suph2C

Zff�sg

h.s � f / d� D suph2C

Zh.s � f / d� D Gs � inf

h2C

Zhf d�

D Gs �

Zgf d� D Gs �

Zff <sg

f d� � cs�.ff D sg/:

Proof. It suffices to show that

8h 2 C W

Z.g � h/f d� � 0:

WriteZ.g � h/f d� D

� Zff <sg

C

ZffDsg

C

Zff >sg

�.g � h/f d�

� s

Zff <sg

.1 � h/ d�C s

ZffDsg

.c � h/ d� �

Zff >sg

hf d�

� s�.ff < sg/C sc�.ff D sg/ � s

Zff�sg

h d� � s

Zff >sg

h d�

D sG � sG D 0

(2.2) Example. Take X D N, and � to be the counting measure. Then problem becomesfinding fg˛g 2 C , such that X

˛2N

g˛f˛ D inffh˛g2C

X˛2N

h˛f˛;

9

whereC D

nfg˛g � R W 0 � g˛ � 1;

X˛2N

g˛ D Go:

The minimizer fg˛g is given by

g˛ D

8<:1; iff˛ < sc; iff˛ D s0; otherwise

wheres D supft 2 R W jf˛ 2 N W f˛ < tgj � Gg:

Therefore X˛2Nf˛<s

f˛ CX˛2Nf˛Ds

c D inffh˛g2C

X˛2N

h˛f˛:

2.2 Lp spaces(2.3) Example. For what exponents k > 0 does k�k�k belong to Lp? Using the coarea formulaZ

B.0;r2/XB.0;r1/

1

kxkkpdx D

Z r2

r1

Z@B.0;r/

1

rkpd� dr

D Cn

Z r2

r1

rn�1�kp dr D Cn

(log r2 � log r1; if k D n=prn�kp2 � r

n�kp1 ; if k ¤ n=p

We deduce that for k < n=p, k � k�k has p-integrable singularity at 0, whereas for k > n=p,it is p-integrable at 1. Therefore on a bounded domain � � Rn containing 0, we havek � k�k 2 Lp.�/ if and only if k < n=p.

(2.9) Riesz’s selection lemma. Let ff˛g � Lp be Cauchy, 1 � p < 1. Then there exists asubsequence ffˇ g � ff˛g, functions f; g 2 Lp, such that jfˇ j � g a.e., and fˇ ! f a.e.

(2.10) Theorem. Let .X;F ; �/ be a measure space, f W X �! R measurable. Suppose thatthere exists an exponent 0 < p <1, such that kf kp < C1. Then

limq!1

kf kq D kf k1:

(2.11) Lemma. Let 0 < p0 < p1 � 1, and f 2 Lp0.X/ \ Lp1.X/. Then f 2 Lp.X/ forp 2 Œp0; p1�, and

8� 2 .0; 1/ W kf kp� � kf k1��p0kf k�p1;

where1

p�D1 � �

p0C�

p1:

10

2.3 Maximal Functions(2.12) Calderon–Zygmund decomposition. Let f 2 L1.Rn/, � > 0. Then there exists afamily of cubes fQ˛g � Rn with disjoint interiors, such that

(1) jf j � � a.e. on .SQ˛/

c .

(2) For all ˛,

� <1

jQ˛j

ZQ˛

jf j d� < 2n�

Furthermore, there exists functions g and b, such that f D g C b, called the ”good part” and”bad part” of f , respectively that satisfy

(1) jgj � 2n� and kgk1 � kf k1

(2) b DPb˛, where supp b˛ � Q˛, kb˛k1 � 2

RQ˛jf j d� and

Rb˛ d� D 0.

2.4 Modes of Convergence(2.13) Definition. Let .X;F ; �/ be a measure space, ff˛g and f measurable functions. Thesequence ff˛g is said to converge in measure to f , if

8" > 0 W �.fjf˛ � f j � "g/! 0:

(2.14) Definition. Let .X;F ; �/ be a measure space, ff˛g and f measurable functions. Thesequence ff˛g is said to converge quasiuniformly to f , if

8" > 0 9N 2 F ; �.N / < " W f˛ ! f uniformly on N c:

(2.15) Remark. As in the classical case, quasiuniform convergence implies almost everywhereconvergence. To see this, choose "ˇ D 1=ˇ in the defnition, then f˛ ! f uniformly on N c

ˇ,

where �.Nˇ / < 1=ˇ. Then N DTˇ�1Nˇ is a null set, and f˛ ! f pointwise on N c .

(2.16) Egoroff’s theorem. Let .X;F ; �/ be a measure space with �.X/ < C1, ff˛g and fmeasurable functions. If f˛ ! f a.e., then f˛ ! f quasiuniformly.

Proof. By changing f on a set of measure zero, we may assume that f˛ ! f everywhere. Let

8˛; ˇ 2 N W E˛;ˇ D[��˛

˚jf� � f j � 1=ˇ

2 F :

Then obviously E˛C1;ˇ � E˛;ˇ , and due to the convergence f˛ ! f ,\˛�0

E˛;ˇ D ;:

From elementary measure theory (using the hypothesis that�.X/ < C1), we have�.E˛;ˇ /!0 as ˛ ! 1. Hence for every " > 0, ˇ 2 N there exists an index ˛ˇ 2 N, such that�.E˛ˇ;ˇ / < "2�ˇ . Let N D

Sˇ2NE˛ˇ;ˇ , then �.N/ < " and for every x 2 N c , we have

jf˛.x/ � f .x/j < 1=ˇ for all ˇ and ˛ � ˛ˇ .

11

(2.17) Lusin’s theorem.

From Bauer. Let .X;F ; �/ be a measure space, 1 � p <1.

(2.18) Lemma. Let f W X �! R be measurable. Then f 2 L1.X/ if and only if

8" > 0 9g 2 L1.X/; g � 0 W

Zfjf j�gg

jf j d� � ": (1)

Proof. If f is summable, take g D 2jf j, then fjf j � gg D ff D 0g [ fjf j D 1g. The levelset N D fjf j D 1g is �-measureable, because f is measurable. Furthermore, a�N � jf jon N for every a 2 R. Therefore a�.N/ �

Rjf j d� < 1, from which �.N/ D 0 follows.

Hence the integral in (1) is zero.Conversely, by splitting the integral into a ”large” and a ”small” part,Z

jf j d� D

Zfjf j�gg

jf j d�C

Zfjf j�gg

jf j d� �

Zg d�C " <1

We conclude that the property embodied in (1) becomes useful when applies to a family ofmeasurable functions in a uniform fashion.

(2.19) Definition. Let E be a family of measurable functions X �! R. Then E is calledequiintegrable if

8" > 0 9g 2 L1.X/; g � 0 8f 2 E W

Zfjf j�gg

jf j d� � ":

(2.20) Remark. Let h 2 L1.X/, h � g, then obviously fjf j � hg � fjf j � gg, hence thefuncion h will also do in the definition. It follows that a finite union of equiintegrable sets fEkgwith respective bounds fgkg is equiintegrable, since g D maxfg1; : : : ; gng will do. Hence, byreferring to the lemma, any finite set of summable functions is equiintegrable.(2.21) Remark. Lebesgue’s theorem actually postulates that the sequence is equiintegrable.To see this, let E be a set of measurable functions, and suppose that there exists a summablemajorant of E , that is, a function g 2 L1.X/, g � 0, such that jf j � g for all f 2 E , a.e. onX . Then E 0 D fjf j W f 2 Eg is equiintegrable, since h D 2g is an adequate uniform bound,

8f 2 E W

Zfjf j�hg

jf j d� �

Zfg�hg

g d� D

ZfgDC1g

g d� D 0:

(2.22) Theorem. A set E of measurable functions is eqiintegrable if and only if

(1) E is uniformly bounded, i.e.

supf 2E

Zjf j d� < C1;

(2) 8" > 0 9ı > 0 9h 2 L1.X/; h � 0 such that

8E 2 F 8f 2 E W

ZE

h d� < ı H)

ZE

jf j d� < ":

12

(2.23) Remark. There are different definitions of equi-integrability in the literature. Ours isa ”functional” version, but the following called ”uniform absolute continuity of integrals” issomewhat more common,

8" > 0 9ı > 0 8E 2 F 8f 2 E W �.E/ < ı H)

ZE

jf j d� < " .�/

Assume�.X/ < C1�.X/ < C1�.X/ < C1. We see that by taking h � 1 in (2) of the theorem above, our definitionimplies .�/. Conversely, let E D fjf j � 1g, and g � 1. Then

Rfjf j�1g

jf j < ".(2.24) Remark. Another definition of equiintegrability used frequently in probability contextsis

limC!0

supf 2E

Zfjf j�C g

jf j d� D 0:

It can be shown that if �.X/ < C1, then this is also equivalent to our definition.

(2.25) Vitali convergence theorem. Let .X;F ; �/ be a measure space, assume that �.X/ <C1 and let 1 � p <1. If ff˛g � Lp.X/ and f 2 Lp.X/, then

kf˛ � f kp ! 0 () f˛ ! f .in measure/; and fjf˛jpg is equiintegrable:

(2.26) Theorem. Let .X;F ; �/ be a measure space, assume that �.X/ < C1 and let 1 � r <p <1. If ff˛g � Lp.X/ is bounded, and f˛ ! f in measure, then kf˛ � f kr ! 0.

(2.27) Vitali–Hahn–Saks theorem. Let .X;F ; �/ be a measure space, and assume that�.X/ < C1. If ff˛g � L1.X/ and f 2 L1.X/, then

kf˛ � f k1 ! 0 () f˛ ! f .in measure/; and f˛ * f .in �.L1; L1//:

(2.28) de la Valee Poussin theorem. Let .X;F ; �/ be a measure space, and assume that�.X/ < C1. Then E � L1.X/ is equiintegrable if and only if 9G W Œ0;C1/ �! Œ0;C1/

monotone increasing, such that

limt!C1

G.t/

tD C1; and sup

f 2E

ZG ı jf j d� < C1:

Moreover, G can chosen to be convex.

2.5 Lattice Structure

2.6 Caratheodory’s Construction(2.29) Definition. Let X be a nonempty set. A nonnegative set function ' W P .X/ �!

Œ0;C1/ is called an outer measure if

(1) '.;/ D 0

(2) 8Q1;Q2 2 P .X/ W Q1 � Q2 H) '.Q1/ � '.Q2/

(3) If J is a countable index set, and fQ˛ W ˛ 2 J g � P .X/, then

'� [˛2J

Q˛

��

X˛2J

'.Q˛/:

13

(2.30) Definition. A subset Q 2 P .X/ is Caratheodory-measurable if

8A 2 P .X/ W '.A/ D '.A \Q/C '.A \Qc/:

For every set A 2 P .X/ we introduce the restriction 'bA via

8B 2 P .X/ W .'bA/.B/ D '.A \ B/;

which is an outer measure on P .A/. Hence Caratheodory’s criterion may be written as

8A 2 P .X/ W .'bA/.X/ D .'bA/.Q/C .'bA/.Qc/:

(2.31) Charatheodory’s theorem. The family Q � P .X/ of Caratheodory-measurable setforms a � -algebra, on which the outer measure ' is � -additive.

In summary, Caratheodory’s theorem provides a general process for constructing measurespaces. There are various ways of defining outer measures, most of them are geometricallymotivated. Next, we define a quite general framework for this task.

Defining outer measures. The following inputs are needed for a covering-type outer mea-sure.

(1) A covering family of sets C � P .X/, for which ; 2 C .

(2) A nonegative set function � W A �! Œ0;C1/ on the covering family.

Next define the system of all possible countable coverings with A,

8A 2 P .X/ W C.A/ DnC 0 W J countable; C 0 D fC˛ W ˛ 2 J g; A �

[˛2J

C˛

o:

Using the set function �, we may assign a quantity for each contable covering, and thus foreach set,

8A 2 P .X/ W ��.A/ D

8<:

infn XC2C 0

�.C/ W C 0 2 C.A/o; if C.A/ ¤ ;

C1; if C.A/ D ;

2.7 Lebesgue and Hausdorff Measures

2.8 Borel and Radon MeasuresAs mentioned before, given a Hausdorff topology T on X , the � -algebra generated by T ,denoted as B.X/ is called the Borel � -algbera (we won’t be using multiple topologies on X ,so the notation B.X/ is just fine). Note that the complements of open sets in T are included inB.X/, it follows that every compact set K � X is a Borel set, since it is closed.

Borel and Radon measures. A measure � given on a Borel � -algebra B.X/ is called aBorel measure, if

8K � X compact W �.K/ < C1:

Furthermore, a measure � is said to be

(1) Inner regular (or tight) on A � X if

�.A/ D supf�.K/ W K � A compactg:

14

(2) Outer regular on A � X if

�.A/ D inff�.U / W U � A openg:

(3) Borel regular if

8A � X 9E 2 B.X/ W A � E; �.A/ D �.E/:

(4) Locally finite if every point in X has an open neighborhood U , such that�.U / < C1.

(5) Radon measure if it is Borel, and Borel regular.

(2.32) Remark. Under our standing assumption that X is a locally compact, separable metricspace, a Radon measure becomes inner regular on every set and outer regular on every Borelset.

We are mostly interested in Radon measures since they constitute the dual of compactlysupported continuous functions.

2.9 Young Measures

3 Analysis in Vector SpacesAs a general ruleX� denotes the topological dual of the vector spaceX ifX has some topology,otherwise X� is the algebraic dual.

3.1 Hyperplanes. The Hahn–Banach Theorem and its CorollariesLet X be a real vector space. A subspace H � X is called a (homogeneous) hyperplane if ithas codimension one, i.e. dim.X=H/ D 1. A somewhat more elementary, equivalent definitionis that H is a hyperplane if (i) a nontrivial subspace (i.e. ¤ f0g and ¤ X ), and (ii) whenevery 2 H c , we have spanfyg ˚H D X .

An affine (or inhomogeneous) hyperplane A is an element of X=H ; if ' 2 X�, such thatker' D H , then the affine hyperplane is given the c-level set of ', i.e.

9c 2 R W A D fx 2 X W .'; x/ D cg:

This is because for a fixed y 62 H , A D cy C ker' (i.e. a coset) for a suitable c 2 R, by thedefinition of the quotient space.

(3.1) Representation theorem for hyperplanes. Homogeneous hyperplanes are precisely thekernels of nonzero linear functionals on X .

Proof. 1. Let 0 ¤ ' 2 X�, then ker' D '�1.0/ � X is a nontrivial subspace. If x 2 X andy 2 .ker'/c , then

x �.'; x/

.'; y/y 2 ker';

in other words dim.X= ker'/ D 1.2. Now suppose that H � X is a hyperplane, that is, H is a nontrivial subspace and

dim.X=H/ D 1. Then for a fixed y 2 H c , every element x 2 X can be written uniquely inthe form x D ty C z, where z 2 H and t 2 R. Now let .'; x/ D t for every x 2 X . Note thatusing the uniqeness of the decomposition, we have ker' D H .

15

(3.2) Proposition. Let X be a normed space.

(1) A hyperplane is closed if and only if its representing functional ' is continuous.

(2) A hyperplane is either closed or dense.

(3.3) Proposition. Let X be a Banach space. Then

8' 2 X 0 8x 2 X W j.'; x/j D k'kd.x; ker'/:

Proof. First we prove the ”�” part. Let ' 2 X 0, x 2 X and z 2 ker', then

j.'; x/j D j.'; x � z/j � k'kkx � zk;

or taking infium in z 2 ker',

j.'; x/j � k'k inffkx � zk I z 2 ker'g D k'kd.x; ker'/:

Second prove the converse inequality. Let y 2 .ker'/c , then x � y .';x/.';y/

2 ker' as before.Now

d.x; ker'/ � x � �x � .'; x/

.'; y/y� D j.'; x/j

j.'; y/jkyk �

j.'; x/j

j.'; y/j

j.'; y/

k'kDj.'; x/

k'k

(3.4) Remark. Suppose that X D Rn and then X 0 D Rn and the situation can be interpretedgeometrically as follows. The vector y 62 ker' is a normal to the hyperplane ker', and thelinear map

P W Rn �! Rn; P x D x � y.'; x/

.'; y/

is an oblique projection (i.e. P 2 D P ) onto that hyperplane. If ' and y are proportional, thenit is an orthogonal projection.

(3.5) Hahn–Banach theorem. Let X be an real vector space. Let p W X �! R be a sublinearfunctional, i.e.

(i) 8� > 0 8x 2 X W p.�x/ D �p.x/

(ii) 8x; y 2 X W p.x C y/ � p.x/C p.y/

and ' W Y �! R is a linear functional defined on some subspace Y � X . Suppose that ' isdominated by p on Y , by which we mean

8y 2 Y W .'; y/ � p.y/:

Statement. There exists an extension Q' 2 X� of ' to the whole space, dominated on all of Xby p, i.e.

(i) 8y 2 Y W .e'; y/ D .'; y/(ii) 8x 2 X W .e'; x/ � p.x/

16

(3.6) Corollary. Let X be a normed space, Y � X a subspace and x0 2 Y c a fixed point. Thenthere exists ' 2 X 0, such that

(1) k'k D 1

(2) ' � 0 on Y

(3) .'; x0/ D d.x0; Y /

Proof. Let Y1 D Y ˚ spanfxg, and define '1 2 Y �1 via the instruction

8y1 2 Y 8� 2 R W y1 D y C �x H) .'1; y1/ D �d.x; Y /:

The only property not immediate is that '1 continuous and k'1k D 1. Since d.x; Y / � kx�ykfor every y 2 Y , we have for every y1 2 Y1, � 2 R and y 2 Y , such that y1 D y C �x,

j.'1; y1/j D j.'1; y C �x/j D j�jd.x; Y / � j�j x C y

�

D k�x C yk D ky1k;so k'1k � 1. Now via the Hyperplane theorem

d.x; Y / D .'1; x/ D k'1kd.x; ker'1/ D k'1kd.x; Y /;

so k'1k D 1. The functional '1 is dominated by the subadditive functional y1 7! k'1kky1k,therefore by the Hahn–Banach theorem can be extended to a functional ' 2 X�, that is stilldominated by the said subadditive functional. In other words k'k D 1.Applying the corollary to Y D f0g, we have

(3.7) Corollary. Let X be a normed space, x0 ¤ 0 a fixed point. Then there exists ' 2 X 0,such that

(1) k'k D 1

(2) .'; x0/ D kx0k

3.2 Topological Dual Space of a Normed SpaceWe begin by summarizing the concept of the topological (or continuous) dual space X 0 of anormed space X . The topological dual space is defined as X 0 ´ B.X;R/, i.e. the space ofbounded linear functionals on X , a subspace of the algebraic dual X�. It is easy to show thatX 0 is a Banach space with the operator norm

8' 2 X 0 W k'k D supfj.'; x/j W kxk D 1g:

We will be using the duality pairing .�; �/ W X 0 � X �! R (a continuous bilinear func-tional) notation originating from Dirac’s bra-ket. This notation is ”symmetric” in the sense thatinterchange the arguments freely.(3.1) Example. LetX D Rn with the Euclidean norm, and consider the functionals fykg � X 0,defined as .yk; x/ D �k for 1 � k � n, where x D

Pk �kxk and fxkg � X is the standard

basis. Then the yk’s are bounded, since j.yk; x/j D j�kj � kxk and j.yk; xk/j D kxkk, sokykk D 1. Note that .yk; x`/ D ık` and therefore fykg forms a basis in X 0. The linear mapˆ W X 0 �! X defined as

8 2 X 0 W ˆ DXk

kxk;

17

where DPk kyk, so we get . ; x/ D .ˆ ; x/. It is easy to check that ˆ is an isometric

isomorphism, that is, kˆ k D k k and surjective. This is the situation in infinite dimensionalHilbert spaces too, but the above proof is not applicable. Can you see why? Also, can you seethe geometric action of ˆ?(3.2) Example. The Riesz representation theorem for Hilbert spaces gives a characterizationof the dual space. Let H be a (real) Hilbert space. The theorem says that the action of everybounded linear functional 2 H 0 can be expressed as . ; x/ D .ˆ ; x/ where ˆ W H 0 �!H is an isometric isomorphism and .�; �/ denotes both the inner product and the duality pairing,depending on the arguments. In summary, the dual space of a Hilbert space is identical to theoriginal up to an isomorphism.(3.3) Example. The Lebesgue spaces Lp.X;F ; �/ furnish our a prime examples for 1 � p �1. We have .Lp/0 Š Lp

0

, where p�1Cp0�1 D 1 and p ¤ 1 and p ¤1. Can you check thatthe linear map ˆ W Lp �! Lp

0

defined by the instruction

8f 2 Lp 8g 2 Lp0

W . f; g/ D

ZX

fg d�

is an isometric isomorphism for 1 < p <1, so ˆ.Lp/ D Lp0

? It is somewhat surprising howsimple this is. It is harder to show that ˆ is also a L1 �! .L1/0 isometric isomorphism.(3.4) Example. Dual spaces of compactly supported continuous functions C0.K/ are consid-ered here, where K is a locally compact Hausdorff space. The Riesz–Markov representationtheorem states that for every ' 2 C0.K/0, there exists a unique, regular (signed) Borel measure� on K, such that

8f 2 C0.K/ W .'; f / D

ZK

f d�:

Positivity of ' is equivalent to the nonegativity of �. The norm of ' is given by the totalvariation of �. That is C0.K/0 ŠM, where M.K/ is the vector lattice of regular signed Borelmeasures on K and the isometric isomorphism is given by

ˆ W M.K/ �! C0.K/0; .ˆ�; f / D

ZK

f d�:

(3.5) Example. The previous example of a dual space is very important, but may be little tooabstract. To get intuition on how it works, consider K D Œ�1; 1�. Since continuous functionshave well-defined pointwise values, we may take .'; f / D f .0/ for f 2 C0.K/. Then theRiesz–Markov representation theorem says that there exists a unique signed Borel measure� 2M.K/ such that

.'; f / D

ZK

f d� and k'k D k�k:

This measure � is of course the Dirac measure concentrated on 0, i.e.

ı0.E/ D

(1; 0 2 E

0; 0 … E

for every Borel set E � Œ�1; 1�. To see this, construct a sequence of step functions ffng, suchthat fn ! f pointwise on Œ�1; 1�, for example as follows:

fn DXjkj�n

f�kn

��Œkn ;

kC1n /:

18

Then ZK

fn dı0 D

(f .0/; n D 2mC 1

f .�n�1/; n D 2m

and Lebesgue’s theorem yields the assertion, since jfnj � kf k.

3.3 Reflexive spacesIterating this procedure of taking duals we get the bidual X 00 D B.X 0;R/. An importantobservation is that there exists a natural injection

i W X �! X 00; 8x 2 X 8' 2 X 0 W .i.x/; '/ D .'; x/:

It follows that i.x/ is bounded on X 0, since

j.i.x/; '/j D j.'; x/j � k'kkxk;

that is ki.x/k � kxk. A conseqence of Hahn–Banach theorem is that for a fixed element x 2 Xthere exists 'x 2 X 0 such that k'xk D 1 and .'x; x/ D kxk. What happens if we feed this 'xto i.x/?

.i.x/; 'x/ D .'x; x/ D kxk:

In other words the supremum in the definition of the operator norm is attained on 'x, thereforeki.x/k D kxk. That is, i W X �! X 00 is an isometry – this justifies injectivity.

An important situation is that when this particular injective isometry, also called the canon-ical embedding, i W X �! X 00 is surjective, in other words maps onto X 00. Then X is calledreflexive. Reflexive spaces have particularly nice properties, as we shall see in the sequel.

First of all,X is reflexive if and only ifX 0 is reflexive, proving this involves some notationalconfusion.

3.4 Uniformly Convex SpacesLet’s have look at a famously reflexive class of Banach spaces. The following geometric prop-erty entails some picture, something the reader should envision.

(3.8) Definition. A Banach space X is said to be uniformly convex if

8" > 0 9ı > 0 8x; y 2 X W kxk � 1; kyk � 1; kx C yk � 2 � ı H) kx � yk < ":

(3.6) Example. Every Hilbert space is uniformly convex – use the parallelogram law.An important pair of inequalities follow, from which the uniform convexity of Lp spaces canbe deduced.

(3.9) Hanner’s inequalities. Let f; g 2 Lp.X;F ; �/. If 2 � p <1, then

kf C gkp C kf � gkp � .kf k C kgk/p C jkf k � kgkjp

.kf C gk C kf � gk/p C jkf C gk � kf � gkjp � 2p.kf kp C kgkp/

If on the other hand 1 � p � 2, the inequalities are reversed.

(3.10) Remark. For p D 1, Hanner’s inequalities reduce to the reverse and the ordinary triangleinequality, respectively. As p D 2, the theorem states the parallelogram law.

19

(3.11) Proposition. The spaces Lp.X;F ; �/ are uniformly convex for 1 < p <1.Proof. Following Komornik, let " > 0

K Dnx 2 R2 W kxkpp D 2; S.x/ � "

o;

T W K �! R; T .x/ Dkxk

pp

2�

ˇx1 C x22

ˇp;

S W K �! R; S.x/ Dˇx1 � x2

2

ˇp:

Then K is compact, since it is the intersection of the compact set @Bp.0; 21=p/ with the closedset fS.x/ � "g. Therefore we may deduce that T attains its minimum � that is strictly positivedue to j � jp being convex.

Next, we prove

8x 2 R2 W"

2kxkpp � S.x/ H)

�

2kxkpp � T .x/:

First, note that T .ax/ D apT .x/ and S.ax/ D apS.x/ for all a > 0, therefore we may assumekxk

pp D 2. But then the implication is trivial from the definition of �.Now let f; g 2 Lp.X;F ; �/, and suppose that kf kp � 1, kgkp � 1 and

fCg2

pp� 1� ı.

Consider the level set

M D fx 2 X W S.f .x/; g.x// �"

2k.f .x/; g.x//kpp g

Note thatRXk.f; g/k

pp d� � 2, soZ

X

ˇf � g2

ˇpd� D

ZM c

S.f; g/ d�C

ZM

S.f; g/ d�

< "

ZM c

k.f; g/kpp

2d�C

ZM

k.f; g/kpp

2d�

� "C1

�

ZM

T .f; g/ d� D "C1

�

ZM

hk.f; g/k

pp

2�

ˇf C g2

ˇpid�

� "C1

��1 � ı

�D "C

ı

�

In this paragraph we turn our attention to projection. It is well-known that in Hilbert space,projection onto a convex, closed set is unique and has an extremely useful variational charac-terization. But these properties are alse enjoyed in a more general spaces.

(3.12) Definition. Let .X; d/ be a metric space, K � X nonempty. The set-valued map PK WX �! P .K/ is called the metric projection onto K if

8x 2 X W PK.x/ D fy 2 K W d.y; x/ D d.x;K/g:

If PK.x/ ¤ ; for all x 2 X , K is called proximal. If PK.x/ is a singleton set for all x 2 X , Kis called a Chebyshev set.

In other words a Chebyshev set is a proximal set onto which the metric projection is unique.

(3.13) Szokefalvi-Nagy theorem. Let X be a uniformly convex Banach space. Then everyK � X nonempty, convex and closed set is Chebyshev.

20

Proof. Let d D d.x;K/, and fy˛g � K a minimizing sequence: ky˛ � xk ! d . Now let

t˛ D1

ky˛ � xk; and z˛ D t˛.y˛ � x/;

then t˛ ! 1=d and kz˛k ! 1.Next we prove that fz˛g is Cauchy using uniform convexity,

˛ ¤ ˇ W kz˛ C zˇk D kt˛.y˛ � x/C tˇ .yˇ � x/k

D .t˛ C tˇ / t˛

t˛ C tˇ.y˛ � x/C

tˇ

t˛ C tˇ.yˇ � x/

D .t˛ C tˇ /

x � t˛y˛ C tˇyˇt˛ C tˇ„ ƒ‚ …2K

� .t˛ C tˇ /d ! 2:

Since X is complete,K is closed and fz˛g is Cauchy, there exists z 2 X , such that z˛ ! z.Then y˛ D z˛=t˛ C x ! z=d C x D y, and ky � xk D lim ky˛ � xk D lim kz˛=t˛k D d , soy 2 K since K is closed.

As for uniqueness, let y; y 0 2 K, such that kx � yk D kx � y 0k D d . The sequence.y; y 0; y; y 0; : : :/ � K is minimizing, which yields using the uniqeness of limits in Hausdorffspace that it is the constant sequence, i.e. y D y 0.(3.7) Example. In L1 and L1, there can more than one projections, therefore they are notuniformly convex.

Of course what we wish for in every space is a tool similar to the one in Hilbert space.

(3.14) Hilbert space projection theorem. Let H be a Hilbert space, K � H a nonempty,closed and convex set. Then the metric projection onto K, PK W H �! K has the followingproperties.

(1) PK is nonexpansive, i.e. 8x; y 2 H W kPK.x/ � Pk.y/k � kx � yk.

(2) The projection PK.x/ of a point x 2 H is characterized by the variationalinequality

8y 2 K W .x � PK.x/; y � PK.x// � 0:

(3) If, moreover, K is a subspace, then the previous variational inequality reducesto .x � PKx; y/ D 0 (8y 2 K) and PK is a linear operator with kPKk � 1.

(3.15) Theorem. Let X be a uniformly convex Banach space.

(1) If K is a nonempty, convex and closed set, then the map PK is continuous.

As an interesting sidenote, reflexivity is in fact charcaterized by the proximality of convex,closed sets.(3.16) Theorem. Let X be a Banach space.

(1) X is reflexive if and only if every nonempty, convex, closed set is proximal.

(2) IfX is reflexive, thenX is strictly convex if and only if every nonempty, convexand closed set is Chebyshev.

21

Returning to reflexivity, we have the following.

(3.17) Milman–Pettis theorem. Every uniformly convex Banach space is reflexive.

This result is sharp, as there is a reflexive Banach space which is not isomorphic to any uni-formly convex space.

3.5 Strictly Convex Spaces

3.6 Subdifferentials. Duality MappingsFrom Chidume, Deimling, Li, etc.

(3.18) Definition. Let X be a Banach space, K � X convex and j W K �! R [ fC1gconvex, and x 2 K an arbitrary point.

(1) The functional ' 2 X� is called a subgradient of j at x if

8y 2 K W j.y/ � j.x/ � .'; y � x/:

(2) The set @j.x/ � X� is called the subdifferential of j at x if

@j.x/ D f' 2 X� W j.y/ � j.x/ � .'; y � x/ .8y 2 K/g:

(3) The set-valued map @j W X �! P .X/ is called the subdifferential of j .

(3.8) Example. Consider the function j � j W R �! R at x D 0. A subgradient ' 2 @j � j.0/ ischaracterized by

80 ¤ y 2 R W jyj � 'y:

Therefore @j � j.0/ D Œ�1; 1�. In general a piecewise linear real-real function has similar subd-ifferentials at the vertices. At other points, the subdifferential is just a singleton consisting ofthe slope.(3.19) Lemma. Let X be a normed space, j W X �! R convex and Gateux-differentiable.Then

8x; y 2 X W j.y/ � j.x/ � .Dj.x/; y � x/:

Proof. Let jx;y.t/ D j..1 � t /x C ty/ for t 2 Œ0; 1�, and note that jx;y is convex. Fromelementary real analysis, jx;y.1/ � jx;y.0/ � j 0x;y.0/.

(3.20) Proposition. Let j be Gateaux-differentiable. Then @j.x/ D fDj.x/g for all x 2 X .

Proof. Suppose that j W X �! R is Gateaux-differentiable. By the previous lemma, Dj.x/ 2@j.x/ for every x 2 X , since

8y 2 X W j.y/ � j.x/ � .Dj.x/; y � x/:

Now suppose that ' 2 @j.x/. Then for all y 2 X and t 2 R,

j.x C ty/ � j.x/ � t .'; y/:

That is, .Dj.x/; y/ � .'; y/ for all y 2 X . Then for �y, .Dj.x/; y/ � .'; y/, and henceDj.x/ D '.

22

A function � W Œ0;C1/ �! Œ0;C1/ is said to be a gauge function (or weight function),if it is continuous, strictly increasing, �.0/ D 0, and limC1 � D C1. It can be shown that theprimitive of a gauge function, i.e. T .t/ D

R t0� , is convex.

(3.21) Definition. Let X be a normed space, � a gauge function. Then the set-valued mapJ� W X �! P .X/ is called the duality mapping on X , if

8x 2 X W J�.x/ D f' 2 X�W .'; x/ D k'kkxk; where k'k D �.kxk/g

If the gauge function is taken to be �.t/ D t , then J D J� is called the normalized dualitymapping.

(3.22) Proposition. Let X be a Banach space. Then

80 ¤ x 2 X W @kxk D f' 2 X� W k'k D 1; .'; x/ D kxkg: (2)

Proof. By definition,

@kxk D f' 2 X� W kyk � kxk � .'; y � x/ .8y 2 X/g:

Note that if ' is in the right-hand side of (2), then .'; y/ � y, hence

8y 2 X W .'; y � x/ � kyk � kxk;

so ' 2 @kxk. Conversely, suppose that ' 2 @kxk, then

8y 2 X W .'; y/ D .'; .y C x/ � x/ � ky C xk � kxk � kyk;

therefore k'k � 1. Moreover, kxk � .'; x/ so in fact k'k D 1 and .'; x/ D kxk.

3.7 Eigenstructure of Compact OperatorsThroghout this section, we assume that X and Y are complex, separable Banach space. Firstof all, we recall that a linear operator A W X �! Y is said to be compact, if the image of everybounded set is totally bounded. A subset K of a metric space is called totally bounded, if forevery " > 0, there exists a finite set of points fx˛g � K, such thatK �

SB.x˛; "/. Of course,

a compact set is always totally bounded. Let us remind the reader that for a subset of a metricspace, the following equivalence holds

(Lindelof-) compact () sequentially compact () complete and totally bounded:

Furthermore, in a complete metric space we have

compact () closed and totally bounded:

The implication(H follows from the fact that a closed subset of a complete metric space iscomplete. Conversely, a compact set is automatically closed (even in a Hausdorff space), andtotally bouneded because of the previous chain of equivalences.

There are multiple equivalent definition of the compactness of an operator. For example, Ais compact iff the image of a bounded set is precompact, i.e. its closure is compact. A sequentialversion of this goes as follows: A is compact iff for every bounded sequence fx˛g � X , the

23

image fAx˛g � Y contains a convergent subsequence. Yet another popular version is that theimage of the closed unit ball is totally bounded.

It is clear that a totally bounded set K is bounded, hence we obtain that a compact operatoris bounded. This is also evident form the sequential definition of compactness. If X D Y . thenthe set of compact operators K.X/ forms a closed, two-sided ideal in B.X/. If X is a Hilbertspace, finite rank operators are dense in K.X/ – this is called the approximation property.A word of warning: Per Enflo constructed a Banach space in 1973 which does not have theapproximation property.

It follows from Riesz’s lemma, see page 4, that if X is infinite dimensional, then usingthe ideal propery of K.X/, a bounded operator having a bounded inverse cannot possiblybe compact. Moreover using the open mapping theorem, if the range of compact operator isclosed, it is necessarily finite dimensional, i.e. the operator is of finite rank. SinceX is closed, acompact operator A W X �! X cannot be surjective in an infinite-dimensional Banach space.As for injectivity, let us introduce the notation

8� 2 C W E� D ker.A � �I/:

Note that for � ¤ 0AˇE�W E� �! E�

is onto, and E� D .A � �/�1.f0g/ is closed by definition. Since ranAˇE�� ranA and A is of

finite rank, AˇE�

is also of finite rank.A powerful generalization of these trivial considerations is the following.

(3.23) Riesz’s theorem. Let X be a Banach space, A W X �! X a compact operator and0 ¤ � 2 C. Then

ker.A � �I/ D f0g () ran.A � �I/ D X:

In other words, A��I is injective if and only if A��I is surjective. But then either of these isequivalent to A� �I being bijective. The reason this theorem was found is that it was requiredfor the develeopment of spectral theory of compact operators. We shortly remind the reader tosome basics.(3.24) Definition. Let A W X �! X be a bounded linear operator.

(1) �.A/ D f� 2 C W .A � �I/ W X �! X is bijectiveg is called the resolventset.

(2) �.A/ D �.A/c is called the spectrum.

(3) �p.A/ D f� 2 C W 90 ¤ x 2 dom.A/; such that Ax D �xg is called thepoint spectrum.

(4) For � 2 �p.A/, the subspace E� D ker.A��I/ is called the eigenspace corre-sponding to the eigenvalue �, and dimE� is called the (geometric) multiplicityof �.

(5) The algebraic multiplicity of an eigenvalue � 2 �p.A/ is defined as

dim1[�D0

ker.A � �I/�

24

In light of the definition of the resolvent set, if A is compact and � ¤ 0 makes A � �Iinjective, it automatically makes A � �I surjective and vice versa. Since A � �I W X �! X

is bounded and bijective, it is, in fact a homeomorphism, i.e. the resolvent operator R.�;A/ D.A � �I/�1 is bounded. Therefore � 2 �.A/. Riesz’s theorem is so important that even itscontrapositive has a name, and a wide range of applications.

(3.25) Fredholm alternative. Let X be a Banach space, A W X �! X a compact operatorand 0 ¤ � 2 C. Then

either 9x ¤ 0 W Ax � �x D 0 or 8y 2 X 9x 2 X W Ax � �x D y:

Note the similarity of the above eigenvalue equations: the first one is called a homogeneousand the second an inhomogeneous eigenproblem. Therefore it either the case that � ¤ 0 isan eigenvalue, or the problem Ax � �x D y admits a unique (why?) solution x for arbitraryy 2 X . In the latter case, � 62 �.A/, hence .A � �I/�1 2 B.X/. From this, it is easy to seethat the solution x depends Lipschitz-continuously on the input y.(3.26) Remark. In the previous considerations and theorems, we elegantly avoided the case� D 0. As noted before, in infinite dimensions a compact operator cannot have a boundedinverse, hence 0 2 �.A/ always holds. It is possible that 0 is an eigenvalue of a compactoperator. As an example, modify the right shift

R W `2 �! `2; Rx D .0; x1; x2; : : :/;

so that it becomes compact say, by multiplying each element with a sequence tending to 0.This follows from that fact that in a Hilbert space H , A 2 B.H/ is compact if and only ifkAe˛k ! 0 for every orthonormal sequence fe˛g � H .

(3.27) Fundamental theorem of compact operators. LetX be a Banach space,A W X �! X

a compact operator. Then

(1) �.A/ X f0g D �p.A/, and �.A/ is countable.

(2) For every � ¤ � 2 �p.A/, we have E� \E� D f0g, and dimE� <1.

(3) If the set �p.A/ � C is infinite, its the only accumulation point is 0.

Next, let us turn to the study of self-adjoint compact operators. We have the followingfundamental result.

(3.28) Schauder’s theorem. Let X and Y be a Banach spaces, A 2 B.X; Y /. Then A iscompact if and only if A� is compact.

Now let A W X �! X be a compact operator. The inhomogeneous eigenproblem

Ax � �x D y (3)

has intimate connections with its the adjoint (or dual)

A�' � �' D ;

where '; 2 X�. Let cokerT D kerT � be the cokernel of an operator T .

25

(3.29) Theorem. Let X be a Banach space, A W X �! X a compact operator. Then

(1) ran.A � �I/ D coker.A � �I/?

(2) ran.A� � �I/ D ker.A � �I/?

(3) dim ker.A � �I/ D dim coker.A � �I/

(3.30) Remark. Note that the theorem says a lot more than what is known a priori in everynormed space, i.e. kerT D .ranT �/?, and ranT D cokerT ?. Since this only yields

ran.A � �I/ D coker.A � �/?

andker.A � �I/? D ran.A� � �I/?? D ran.A� � �I/:

This is an alternative-type theorem, regarding the pair of eigenproblems it states that theinhomogeneous eigenproblem is solvable if and only if and its adjoint homogeneous eigen-problem is not, and vice versa. Moreover, if we let E�

�D coker.A � �I/ for all � 2 �p.A/,

then (3) yields via the fundamental theorem that the eigenspaces of the adjoint are finite dimen-sional and dimE�

�D dimE�. Finally, note that E�

�D E� if A is self-adjoint.

(3.31) Definition. Let X be a Banach space. An operator A 2 B.X/ is said to be a Fredholmoperator if dim kerA <1, dim cokerA <1 and ranA is closed.

We close the discussion of spectral theory by considering self-adjoint compact operators Aon a real Hilbert space H . It is easy to see that � 2 R and E� ? E� for every � ¤ � 2 �p.A/.It is less trivial that even �.A/ � R is true. Using Gelfand’s spectral radius formula, it can beproved that r.A/ D kAk, so

�.A/ � B.0; kAk/:

Next, suppose that H is separable. The ortogonality of eigenspaces enables us to constructan orthognormal sequence fe˛g � H via the Gram–Schmidt process, but an eigenspace withdimE� > 1 might require one more invokation of the said algorithm, albeit this time, it haltsin a finite number of steps. To avoid complications in indexing, we enumerate �p.A/ withmultiplicities, that is

�p.A/ D f�1; : : : ; �dimE�1;

�dimE�1C1; : : : ; �dimE�1CdimE�2

;

: : :g;

where �1 D : : : D �dimE�1D �1, and �dimE�1C1

D : : : D �dimE�1CdimE�2D �2, etc., where

f�˛g D �p.A/ denotes the usual sequence without multiplicities. Due to the convergence�˛ ! 0, we can arrange matters so that

j�1j > j�2j > : : :

holds.That way, each member of the orthonormal basis fe˛g has a corresponding eigenvalue –

with the possibility that an eigenvalue is used multiple, but finitely many times. In summary,because fe˛g? D f0g, we have the direct sum decomposition

H DM˛�1

spanfe˛g DM

�2�p.A/

E�:

26

Note that �p.A/ is still a set in the second form, so multiple instances doesn’t count. The aboveprocess is called diagonalization, similarly to the finite dimensional case, since in the basisfe˛g, the action of the operator A can be viewed as a ”multiplication by an infinite diagonalmatrix”, i.e.

A D

0BBB@�1

�2�3

: : :

1CCCA :(3.32) Hilbert–Schmidt theorem. Let H be a Hilbert space, A W H �! H a compact, self-adjoint operator. Then

A DX˛�1

�˛e˛ ˝ e˛ DX˛�1

�˛P˛;

where P˛ W H �! E�˛ is the orthogonal projection onto the eigenspace corresponding to �˛.

These decompositions provide an easy way of solving (3). Let � 2 �.A/ be a regular value,then the inhomogeneous eigenvalue problem

Ax � �x D y

is be solved by

x DX˛�1

�˛

�˛ � �e˛; where y D

X˛

�˛e˛:

3.8 Courant–Fischer Minimax PrinciplesIn this section, we turn to the variational characterization of the eigenvalues of a compact,self-adjoint operator on a Hilbert space. Such results where first obtained by Lord Rayleigh,motivated by physical problems. Then, Courant, Fischer and Weyl investigated various eigen-value problems for partial differential equations. The importance of the formulation was im-mediately realized, and various other applications where found in more general contexts. Theelegant treatment of this section is borrowed from Lax.

Rayleigh studied the quadratic functional

R.x/ D.Ax; x/

.x; x/;

nowadays called the Rayleigh quotient corresponding to A. Consider the maximization prob-lem for R on the whole space,

M D supfR.x/ W x ¤ 0g D supf.Ax; x/ W kxk D 1g;

were the second equality is due to the homogenity of R, R.kx/ D R.x/ for all k 2 R. Fromthe Schwarz inequality, we have jM j � kAk < C1.

Recall that the ordering �p.A/ D f�˛g satisfied

j�1j > j�2j > : : :

It is trivial to estimate jM j from below by setting x D e1, where Ae1 D �1e1, to obtainjM j � j�1j.

27

For simplicity, we restrict our attention to positive eigenvalues, for the substitution �Aclearly handles the negative case. Henceforth we use the sequence

�1 � �2 � : : : ;

with multiplicities counted. The previous estimate now reads

M � �1: (4)

We claim that the supremum is in fact attained in S D @B.0; 1/. The functional j W S �!R, defined as j.x/ D .Ax; x/ is weakly continuous, since if x˛ * x in H , then Ax˛ ! Ax,by compactness of A, and

jj.x˛/ � j.x/j D j.Ax˛; x˛/ � .Ax; x/j

� j.Ax˛ � Ax; x˛/j C j.Ax; x˛ � x/j

� kAx˛ � Axkkx˛k C j.Ax; x˛ � x/j ! 0;

because x˛ * x and hence fx˛g is bounded. Let fx˛g � B D B.0; 1/ be a maximizingsequence, i.e. j.x˛/!M . Since fx˛g is bounded, there exists z 2 B , and a subsequence fxˇ g,such that xˇ * z. From the weak continuity of j , we have j.xˇ /! j.z/, hence j.z/ DM . Itstill remains to prove that z 2 S . First, it follows from weak lower semicontinuity of the norm(4.23) (page 38), that kzk � 1. Assume for contradiction that kzk < 1. Due to j.z/ DM > 0,z ¤ 0, hence it makes sense to write y D z=kzk 2 S . But then j.y/ DM=kzk2 > M D j.z/,contradicting that z is a minimizer.

The moral of the above variational argument is that the ”sup” is in fact a ”max”. But thereis still more to squeeze out. Compute the Gateaux derivative of R at z 2 S in the directiony 2 H , as

.DR.z/; y/ Dd

dtR.z C ty/

ˇtD0

:

We have

d

dt

.A.z C ty/; z C ty/

.z C ty; z C ty/Dd

dt

.Az; z/C 2t Re.Az; y/C t2.Ay; y/.z; z/C 2t Re.x; y/C t2.y; y/

;

hence.DR.z/; y/ D 2Re.Az; y/ � 2.Az; z/Re.x; y/ D 2Re.Az �Mz; y/:

This quantity is supposed to be 0, for every choice of y 2 H , because at z 2 S , R is maximal(remember the homogenity). We obtain that z is an eigenvector and M is its eigenvalue:

Az DMz:

Using the fact that M is an eigenvalue, the estimate (4) and the maximality of �1, we haveM D �1. In summary, we have proved the following.

(3.33) Theorem. Let H be a complex Hilbert space, A W H �! H a compact, self-adjointoperator. If �1 denotes the largest positive eigenvalue of A, then

�1 D maxx¤0

R.x/

Furthermore, the vector z1 ¤ 0 on which the maximum is attained is an eigenvector to �1.

28

(3.34) Remark. If H finite dimensional, then it makes sense to talk about the smallest positiveeigenvalue �0. It follows easily that

�0 D minx¤0

R.x/:

What we should realize is that this variational technique is suitable for obtaining more (infact all) eigenvalues of A. We shall alter the set on which the maximization takes place. Letz1 D z be the eigenvector obtained from the previous construction. The subspace

D2 D spanfz1g?

is invariant under A, for if x 2 D2, then .x; z1/ D 0 and .Ax; z1/ D .x; Az1/ D .x; �1z1/ D

�1.x; z1/ D 0, so Ax 2 D2. Hence the restriction

A2 D AˇD2W D2 �! D2;

is again compact and self-adjoint. Applying the theorem to A2, we obtain that the largestpositive eigenvalue �2 of A2 is given by

�2 D maxx2D2

R.x/ D max.z1;x/D0

R.x/;

and an eigenvector z2 2 D2 to �2. Note that we omitted the constraint ”x ¤ 0” for brevity. Itis obvious that

�1 � �2:

The next subspace to maximize on is going to be

D3 D spanfz1; z2g?;

yielding�3 D max

x2D3R.x/ D max

.z1;x/D0

.z2;x/D0

R.x/;

and again, an eigenvector z3 2 D3 to �3.Continuing in this successive manner, maximizing over the decreasing spacesD˛, we arrive

at the formula�˛ D max

x2D˛R.x/ (5)

where D˛ is the subspace orthogonal to the first ˛ � 1 eigenvectors,

D˛ D spanfz1; : : : ; z˛�1g?:

For the dual construction, consider the subspaces spanned by the first ˛ eigenvectors,

F˛ D spanfz1; : : : ; z˛g: (6)

Then take a nonzero vector x 2 F˛ and expand it in terms of the first ˛ eigenvectors,

x DX1�ˇ�˛

�ˇzˇ ;

29

with �1; : : : ; �˛ 2 C. The Rayleigh quotient of x is trivially bounded from below by �˛, since

8x 2 F˛ W R.x/ D.Ax; x/

.x; x/D

X1�ˇ�˛

�ˇ j�ˇ j2

X1�ˇ�˛

j�ˇ j2� �˛; (7)

where we exploited �1 � : : : � �˛ in the last inequality. Hence

infx2F˛

R.x/ � �˛:

But taking x D z˛ yields�˛ � inf

x2F˛R.x/ � �˛;

thereby establishing the relation�˛ D min

x2F˛R.x/: (8)

These are constrained maximization and minimization problems for the functional R. Suchformulas were known to Rayleigh, but it took a couple of decades to realize, one can in fact getrid of the eigenvectors.

(3.35) Courant’s principle. Let H be a complex Hilbert space, A W H �! H a compact,self-adjoint operator and assume that the positive eigenvalues of A are indexed in decreasingorder, i.e.

�1 � �2 � : : :

Then�˛ D min

E2E˛�1maxx2E?

R.x/;

where E˛�1 denotes the family of all .˛ � 1/-dimensional subspaces of H .

Proof. Let E 2 E˛�1 be arbitrary. Since dimF˛ D ˛, due to the linear independence ofeigenvectors, and E? is a strictly larger than F˛, we have

E? \ F˛ ¤ f0g:

Let 0 ¤ w 2 E? \ F˛, thenw D

X1�ˇ�˛

!ˇzˇ ;

for some !1; : : : ; !˛ 2 C. The estimate (7), now reads

8E 2 E˛�1 8w 2 E?\ F˛ W R.w/ � �˛:

Taking supremum and enlarging the corresponding set,

8E 2 E˛�1 W supx2E?

R.x/ � supx2E?\F˛

R.x/ � �˛;

thusinf

E2E˛�1supx2E?

R.x/ � �˛:

30

Note that up to this point we cannot be certain whether the ”inf” and the ”sup” is actuallyattained. But the choice E D F˛�1 makes E? D D˛, so the situation becomes precisely thesame as in (5), and hence

�˛ D maxx2D˛

R.x/ D maxx2F?˛�1

R.x/ � infE2E˛�1

supx2E?

R.x/ � �˛

(3.36) Fischer’s principle. Let H be a complex Hilbert space, A W H �! H a compact,self-adjoint operator and assume that the positive eigenvalues of A are indexed in decreasingorder, i.e.

�1 � �2 � : : :

Then�˛ D max

E2E˛

minx2E

R.x/;

where E˛ denotes the family of all ˛-dimensional subspaces of H .

Proof. Let E 2 E˛ be arbitrary, then E \ D˛ ¤ f0g, since dimD˛ D ˛ � 1. Then, fromequation (5), we have

8E 2 E˛ 80 ¤ y 2 E \D˛ W R.y/ � �˛:

Consequently, taking infimum results in

8E 2 E˛ W infx2E

R.x/ � infx2E\D˛

R.x/ � �˛;

and hencesupE2E˛

infx2E

R.x/ � �˛:

As a finishing touch, take E D F˛ as per (6), then using (8),

�˛ D minx2F˛

R.x/ � supE2E˛

infx2E

R.x/ � �˛

(3.37) Weyl’s theorem. Let H be complex Hilbert space, A;B W H �! H compact, self-adjoint operators and assume that the positive eigenvalues ofA andB are indexed in decreasingorder, i.e.

A W �1 � �2 � : : : ;

B W �1 � �2 � : : :

Statement. If A � B , then �˛ � �˛.

Proof. The relation A � B means, by definition, that .Ax; x/ � .Bx; x/ for all x 2 H . HenceRA.x/ � RB.x/ for x ¤ 0. Then minRA.X/ � minRB.x/, so the theorem follows fromFischer’s principle. But Courant’s principle could have also been applied as well.

31

4 Weak topologies

4.1 Locally Convex SpacesLet J be an index set for the family of seminorms fj � j W 2 �g on an arbitrary vector spaceX , that is, norms without the condition that jxj D 0 implies x D 0, a.k.a the nondegeneracycondition.

How to induce a topology by a given family of seminorms? First of all, define the balls forevery 2 � as usual:

B .a; r/ D fx 2 X W jx � aj < rg;

which are convex due to the triangle inequality. The topology T� is the topology induced bythe subbase

S� D fB .a; r/ � X W 2 �; a 2 X; r > 0g;

in other words open sets U 2 T� are precisely those, for which given x 2 U there exists a finiteindex set �0 � � and positive numbers fr g � RC, such that\

2�0

B .x; r / � U:

This last equivalence is evident from the definition of the subbase of a topology. It is sometimesuseful to reduce the situation to the case of the normed vector space.

Our definition of a locally covnex space is borrowed form Conway – we are only interestedin Hausdorff topologies.

(4.1) Definition. Let X be vector space, fj � j W 2 �g a family of seminorms. If theseparation property \

2�

fx 2 X W jxj D 0g D f0g; (9)

holds, then the topology T� generated by the subbase S� is called the locally convex topolgy.

(4.2) Proposition. The locally convex topology is Hausdorff.Proof. First note that from the nondegeneracy condition of the seminorms in the definition, wehave [

2�

fx 2 X W jxj ¤ 0g D X X f0g:

Let x; y 2 X , x ¤ y. Then jx � yj 0 > " for some 0 2 � and " > 0. Therefore the ballsB 0.x; "=2/ and B 0.y; "=2/ are disjoint neighborhoods of x and y via the triangle inequality.

From the previous defintion, we obtain that a compact set in a locally convex space isclosed.

(4.3) Definition. A subset A � X of a locally convex space is said to be bounded, if everyseminorm j � j is bounded on it, i.e.

8 2 � 9C > 0 8x 2 A W jxj � C :

Since the seminorms are continuous, a compact set is also bounded in the above sense.Note that if j�j D 1, then T� is a locally convex topology if and only if j � j is a norm. Before

discussing the general topological properties of T� , let us have a look at a handful of familiarexamples.

32

(4.1) Example. Let X be an arbitrary set. Then the vector space F.X/ real-valued functionsendowed with the family of seminorms

8x 2 X W jf jx D jf .x/j

is locally convex space. It is the topology of pointwise convergence on X .Regarding compactness in F.X/, a highly interesting fact is that F.X/ also has the Heine–

Borel property.

(4.4) Lemma. A closed, bounded subset K � F.X/ is compact.

Proof. Boundedness of K yields that the sets

K.x/ D ff .x/ 2 R W f 2 Kg .x 2 X/

are bounded. For every x 2 X , let K.x/ � Ix be a compact interval. Then I DQx2X Ix is

compact by Tychonoff’s theorem. But F.X/ DQx2X R, so I � F.X/ and is compact. Since

K � I is closed and F.X/ is Hausdorff, the result follows.(4.2) Example. Let X be a topological space. Then C.X/ endowed with the family of semi-norms

8K � X compact W jf jK D maxfjf .x/j W x 2 Kg

is locally convex space. This is the topology of locally uniform convergence.(4.3) Example. Let 1 < p < 1, 1=p C 1=p0 D 1, and consider the Lebesgue spacesLp.X;F ; �/ and Lp

0

.X;F ; �/. We may define a locally convex topology on Lp with thefamily of seminorms

8g 2 Lp0

8f 2 Lp W jf jg Dˇ Z�

fg d�ˇ:

We shall denote this topology with the symbol �.Lp; Lp0

/.(4.4) Example. During the construction of the Lebesgue space the Lp norm starts it life as aseminorm, and it becomes a norm after factorization of the space with respect to the subspaceof functions vanishing a.e.(4.5) Remark. A theorem of Kolmogorov states that a topological vector space is locally con-vex if and only if every neighborhood of 0 contains a convex neighborhood of zero. The readerwill find it quite surprising that such a humble requirement results in a very confortable environ-ment to work in, compared to general topological vector spaces. We won’t discuss topologicalvector spaces here, as it would involve lots of definitions and guiding counterexamples (simi-larly to set-theoretic topology). For a rather axiomatic introduction, see Rudin.

The the resulting locally convex topology is not metrizable in general. Therefore the use ofnets instead of sequences is natural for studying locally convex topologies.

4.2 Separation of Convex SetsLet us turn to the study of convex sets in locally convex spaces. First, two corollaries (seeEkeland) of the Hahn–Banach theorem that hold in general topological vector spaces.

33

(4.6) Definition. Let X be a vector space, A;B � X subsets. An affine hyperplane A � X

given by A D fx 2 X W .'; x/ D cg for some ' 2 X� and c 2 R is said to separate A and Bif the closed half-spaces

fx 2 X W .'; x/ � cg; fx 2 X W .'; x/ � cg;

each contain exactly one of them. If the open half-spaces were used insted, A is said to strictlyseparate A and B .

(4.7) Mazur’s separation theorem. Let X be a topological vector space and

(1) A � X an open convex set,

(2) Y � X an affine subspace,

(3) Y \ A D ;.

Statement. There exists a closed affine hyperplane H , such that Y � H and H \ A D ;.

(4.8) Corollary. Let X be a topological vector space, K � X a convex set, intK ¤ ;. Thenfor every x 2 @K, there exists an affine hyperplane H 3 x, such thatK is entirely contained inone of the closed half-spaces of H (x is a supporting point, H is a corresponding supportinghyperplane).

(4.9) Eidelheit’s separation theorem. Let X be a locally convex space and

(1) A � X an open convex set,

(2) K � X a convex set,

(3) K \ A D ;,

Statement. There exists a closed affine hyperplane H that separates A and K. Equivalently,there exists ' 2 X� and c 2 R, such that

8x 2 A 8y 2 K W .'; x/ < c � .'; y/:

(4.10) Tukey–Klee separation theorem. Let X be a locally convex space and

(1) C � X a closed convex set,

(2) K � X a compact convex set,

(3) C \K D ;,

Statement. There exists a closed affine hyperplane H that strictly separates C and K. Equiv-alently, there exists ' 2 X� and c1; c2 2 R, such that

8x 2 C 8y 2 K W .'; x/ � c1 < c2 � .'; y/:

(4.11) Corollary. Let X be a locally convex space, K � X a closed convex set. Then K is theintersection of the closed hyperplanes containing K.

34

Finally, let us focus our attention on the extension of continuous linear functionals on alocally convex space X . Note that we do not have the induced norm topology on X�, butcontinuity on X may be transferred to X�.

(4.12) Definition. Let X be a locally convex space X . The topological dual X 0 � X� of Xcontains linear functionals continuous in the locally compact topology of X .

(4.13) Theorem. Let X be a locally convex space, Y � X a subspace. A continuous linearfunctional ' on Y may be extended to a continuous linear functionale' on the whole space X .

There is a natural analogy of the orthogonal complement to vector spaces. Here we considercomplementation in the topological dual X 0.

(4.14) Definition. Let X be a locally convex space, A � X and B � X 0 arbitrary subsets. Thesubspaces

A? D f' 2 X 0 W .'; x/ D 0 .8x 2 A/g

andB? D fx 2 X W .'; x/ D 0 .8' 2 X 0/g

are called the annihilator (or somewhat misleadingly the orthogonal complement) of A and B ,respectively.

(4.15) Proposition. Let X be a locally convex space, A � X and Y � X a subspace. Then

(1) span.A/ D A??.

(2) If A? D f0g, then span.A/ D X .

(3) If Y ? D f0g, then Y D X .

4.3 Weak and Weak-* Topologies as Product TopologiesLet us now focus on the relation between the vector space X and its algebraic dual X�.

(4.16) Definition. Let X be a locally convex space. The topology induced by the family ofseminorms

fj � j' W X �! R W jxj' D j'.x/j .' 2 X�/ g

is called the weak topology on X and denoted by �.X;X�/, and the resulting locally convexspace by X� .

(4.17) Definition. Let X be a locally convex space. The topology induced by the family ofseminorms

fj � jx W X��! R W j'jx D j'.x/j .x 2 X/ g

is called the weak-* topology on X� and denoted by �.X�; X/ and the resulting locally convexspace by X��� .

It is important to realize that since we in a locally convex space, it makes perfect sense totalk about the topological dual, induced by the locally convex topology.

35

Product topology. First a little reminder on the product topology as per Munkres. Let J bean index set, fX˛ W ˛ 2 J g and X D

S˛2J X˛. The cartesian product of fX˛g is defined as

�2J

X˛ D ff W J �! X W f .˛/ 2 X˛ .8˛ 2 J /g:

For every ˇ 2 J , define the (canonical) projection mappings �ˇ W �˛2J X˛ �! Xˇ as�ˇ .f / D f .ˇ/ for every f 2�˛2J X˛.

(4.18) Definition. Let T be the topology generated by the subbasis S DSˇ2J Sˇ , where

Sˇ D f��1ˇ .Uˇ / W Uˇ open in Xˇ g;

and are called open cylinders. In other words T is made up of all unions of cylinder sets,i.e. finite intersections of elements of various Sˇ . The resulting space is is designated by thenotation

Q˛2J X˛.

Since the product topology is generated by the preimages of open sets of Xˇ via �ˇ , it makesevery map �ˇ W �˛2J X˛ �! Xˇ continuous and due to the minimal property of generation,it is also the smallest topology on�˛2J X˛ with this property.

(4.19) Theorem. The product topology is the weakest topology that makes every canonicalprojection map continuous.

The following easy proposition characterizes continuity in the product space in terms ofcontinuity of coordinate functions.(4.20) Proposition. Let Y , fX˛ W J g be topological spaces. A mapping f W Y �!

Q˛2J X˛

is continuous if and only if 8˛ 2 J W f˛ D �˛ ı f W Y �! X˛ is continuous.The weak and weak-* topology as relative product topology. Let X be a locally convex

space, and let RX denote the set of functions X �! R; this is exactly the same as the cartesianproduct �x2X R D RX . The topological dual X� is just a subset of RX . Similarly, we have�'2X� R D RX

�

.(Weak-* topology.) Define a family of seminorms on X� with the instruction

8x 2 X W j � jx W X��! R; 8' 2 X� W jf jx D jf .x/j;

i.e. evaluation at x 2 X , used in the definition of the weak-* topology are really just the pro-jection mappings defined on the cartesian product RX . Consider the product topology

Qx2X R.

Since X� � RX , we may endow X� with the subspace (or relative) topology inherited fromQx2X R, which is by definition generated by the open cylinders

8x 2 X W Sx D f' 2 X�W j' � jx < " .8" > 0; 8 2 X

�/g

D fBx. ; "/ W 2 X�; " > 0g;

where the ball Bx. ; "/ is defined simply as

Bx. ; "/ D f' 2 X�W j' � jx < "g:

Note that S DSx2X Sx is precisely the subbase S� given in the definition of the locally

convex topology, in other words we are talking about the weak-* topology. In summary, a setU � X� is open in the weak-* topology �.X�; X/ if

8' 2 U 9" > 0 9A � X finite W\a2A

Ba.'; "/ � U:

36

(Weak topology.) Define a family of seminorms on X with the instruction

8' 2 X� W j � j' W X �! R; 8x 2 X W jxj' D j'.x/j;

i.e. evaluation of ', are the projection mappings of RX�

� X��. Via the natural imbedding

i W X �! X��; 8x 2 X ' 2 X� W .i.x/; '/ D .'; x/;

we let bX D i.X/ � X��. Now i W X �! bX is an isomorphism and a homeomorphism (checkthis using nets!). Since X�� is the topological dual of X�, it may be endowed with the weak-*topology �.X��; X�/. The linear subspace bX � X�� inherits this relative weak-* topology,denoted as �.bX;X��/. We now check that by pulling back this topology to X , we get nothingbut the weak topology on X , �.X;X�/. The relative product topology on bX , inherited fromQ'2X� R is generated by the open cylinders

8' 2 X� W S' D f� 2 bX W j� � �j' < " .8" > 0; 8� 2 bX/gD f� 2 bX W j.� � �; '/j < " .8" > 0; 8� 2 bX/g:

Since for every �; � 2 bX , we have i.x/ D � and i.y/ D � for some x; y 2 X , then j.���; '/j Dj.i.x � y/; '/j D j.'; x � y/j for all ' 2 X�. Thus, after our excursion to the bidual X��, wegot exactly the weak topology �.X;X�/.

Again, an open set U � X in the weak topology �.X;X�/ is then characterized by

8x 2 U 9" > 0 9ˆ � X� finite W\'2ˆ

B'.x; "/ � U;

whereB'.x; "/ D fy 2 X W jx � yj' < "g:

Relations between the weak topologies. As noted before, ' 2 X 0 iff for every net fx˛gand x 2 X , such that x˛ ! x, we have .'; x˛/! .'; x/.

The weak topology X� is coarser than the original locally convex topology, in symbols

�.X;X�/ � �.X/;

since the imbedding map X �! X� is continuous. To see this, let fx˛g � X be a net withx˛ ! 0 in X . Then .'; x˛/! 0 for every ' 2 X 0 or, x˛ ! 0 in X� .

In summary, the weak topology �.X;X�/ is the coarsest topology that renders the elementsof X� (i.e. continuous linear functionals, as per the locally convex topology of X ) continuous,and we get nothing new: �.X;X�/� D �.X/�.

Dually, the weak-* topology is coarser than the original dual topology onX�. Furthermore,we have the relations

�.X;X�/ � �.X�; X��/ � �.X/�;

where the first inclusion is due to the fact that i W X �! X�� is an imbedding. Finally, it maybe shown that �.X�; X��/� D �.X/.

4.4 Weak and Weak-* ConvergenceThe previous considerations contained an important concept, which we shall record in a defini-tion. Note the use of nets instead of sequences – this is because real (i.e. not degenerate) weakand weak-* topologies are never metrizable. However, we shall see later that sequences are infact adequate to describe these topologies, but these observations are anything but trivial.

37

(4.21) Definition. Let X be a locally convex space.

(1) A net fx˛g � X is said to be weakly convergent to x 2 X , if

8' 2 X� W .'; x˛/! .'; x/:

Weak convergence is usually denoted as x˛ * x or x˛ ! x in �.X;X�/.

(2) A net f'˛g � X� is said to be weakly-* convergent to ' 2 X�, if

8x 2 X W .'˛; x/! .'; x/:

Weak-* convergence is usually denoted as '˛�

* ' or '˛ ! ' in �.X�; X/.

(4.22) Proposition. Let X be a normed space, fx˛g; fy˛g � X and f'˛g � X�.

(1) A net can have at most one weak limit.

(2) If x˛ * x, then xˇ * x for every subnet fxˇ g � fx˛g.

(3) If x 2 X and .'; x/ D 0 for every ' 2 X�, then x D 0 (separation property).

(4) Weak convergence is linear in the following sense. If x˛ * x, y˛ * y and f�˛g; f�˛g �R, �˛ ! �, �˛ ! �, then

�˛x˛ C �˛y˛ * �x C �y:

(5) We have the following composition properties.

x˛ * x and '˛ ! ' H) '˛.x˛/! '.x/

x˛ ! x and '˛ * ' H) '˛.x˛/! '.x/

(4.23) Weak lower semicontinuity of the norm. Let X be a normed space, fx˛g � X . Then

x˛ * x H) kxk � lim inf kx˛k:

Proof. Let L D lim inf kx˛k, and assume for contradiction that

x 62 B.0;L/:

Apply the Tukey–Klee separation theorem (4.10) (page 34) with K D fxg and C D B.0;L/;hence there exists ' 2 X�, c1; c2 2 R such that

8y 2 B.0;L/ W .'; x/ � c1 < c2 � .'; y/:

Then choosing y D x˛, we see that .'; x˛/ 6! .'; x/.Another proof. Using Corollary (3.7) of the Hahn–Banach theorem, for given x 2 X , thereexists ' 2 X�, such that k'k D 1 and .'; x/ D kxk. Then kxk D .'; x/ D lim.'; x˛/ �lim inf j.'; x˛/j � lim inf kx˛k.

It is worth noting that the Tukey–Klee separation theorem is also a consequnce of the Hahn–Banach theorem, but the above two proofs make different use of it.

38

(4.24) Uniform boundedness principle. Let X be a normed space, fx˛g � X a weakly con-vergent sequence. Then fx˛g is bounded.

Proof. Since fx˛g is weakly convergent, sup˛ j.'; x˛/j <1 for all ' 2 X�. Using the naturalimbedding

i W X �! X��; 8x 2 X ' 2 X� W .i.x/; '/ D .'; x/;

we may deduce that sup˛ j.i.x˛/; '/j < 1 for all ' 2 X�. From the Banach–Steinhaustheorem applied to the sequnce of linear operators fi.x˛/g � X�� defined on the Banach spaceX�, we have sup˛ ki.x˛/k < 1. The imbedding i in a normed space is an isometry (seesection (3.3), page 19), i.e. ki.x˛/k D kx˛k, from which the result follows.More important in the applications is the converse, i.e. when does a bounded sequence possessa weakly convergent subsequence? The next section on weak and weak-* compactness willanswer the question.

The following observation is trivial, but we record it for later purposes anyway.

(4.25) Proposition. If X is a normed space, then

x˛ ! x H) x˛ * x and kx˛k ! kxk:

In a uniformly convex space, the converse implication also holds. A word of warning: inL1 or L1 the weak convergence and the convergence of the norms is not enough to concludeconvergence in the norm.(4.26) Theorem. If X is uniformly convex, then X has the Radon–Riesz property, i.e.

x˛ * x and kx˛k ! kxk H) x˛ ! x:

(4.5) Example. Consider a separable Hilbert space H . Via Riesz’ representation theorem,duality pairing is just the inner product. Let fx˛g � H be an orthonormal basis, then obviouslyfx˛g is bounded, but not Cauchy, since

8˛ < ˇ W kx˛ � xˇk Dp2:

However by Bessel’s inequality, for every x 2 H , we haveX˛

j.x; x˛/j � kxk2:

Therefore .x; x˛/! 0 for all x 2 H , so in fact x˛ * 0.Let’s look at some strange cases, showing that how strong converge can fail hold for weakly

convergent seqeneces. See Lieb and Loss.(4.6) Example. The sequence of characteristic functions f�.˛;˛C1/g � L1.R/ converge weaklyto 0 in L1.R/. To show this, let E D ff � "g, then jEj � "�1kf k1. Hence, for givenf 2 L1.R/, " > 0, we may choose ˛ 2 N, such that E \ .˛; ˛ C 1/ D ;, soZ C1

�1

f�.˛;˛C1/ � ":

This sequence slides to infinity. Again, it is not strongly convergent as we observed earlier, seethe remark after the refined Fatou lemma (2.3), page (2.3).

39

Oscillation. Now let us bring a classical theorem of Fourier analysis into the context ofweak topologies.

(4.27) Riemann–Lebesgue lemma. Let 1 � p; p0 � 1, p0 D .p � 1/=p. Then for everybounded interval I � R, the sequences fsin�nxg and fcos�nxg tend to zero in the topology�.Lp; Lp

0

/, where �n !C1.

This phenomenon is called oscillation and turned out to be highly important for studyingPDE’s. Such wild oscillations cannot be observed in a norm-convergent sequence, that is, thenorm somehow ”damps out” the oscillations. We shall intensely study oscillation using Youngmeasures in a later section.

Concentration. Approximating the ı functional is an archtypical example of this phe-nomenon.

(4.28) Theorem. Let ff˛g � Lp.X;F ; �/ be a bounded sequence (1 < p � 1), i.e.sup˛ kf˛kp <1. If f˛ ! f a.e., then f˛ * f .

Consider the sequence ˛.x/ ´ ˛1=pf .˛x/, where f 2 Lp.R/ and 1 < p < 1. Notethat k ˛kp D kf kp, but ˛ ! 0 a.e., so this bounded sequence is not strongly convergent.But the previous theorem says that ˛ * 0. This phenomenon is named concentration, alsohighly important in modern real analysis.

We mention Schur’s theorem, which says that in `1 the weak coincides with the strongtopology.

(4.29) Kolmogorov’s theorem. LetX be a locally convex space. The following are equivalent.

(1) X is normable.

(2) There exists a bounded neighborhood of 0.

(3) There exists a bounded, nonempty open set.

4.5 Weak and Weak-* CompactnessLet X be a normed space. Let B.X/ and B.X�/ denote the weakly and weakly-* closed unitballs of X and X�, respectively.

(4.30) Banach–Alaoglu theorem. Let X be a normed space. Then B.X�/ is weak-* compact.

Proof. (From Conway.) Let I D Œ�1; 1�, D DQx2B.X/ I D IB.X/. Since I is compact,

Tychonoff’s theorem yields D is compact. Let

G W B.X�/ �! D; G.'/.x/ D .'; x/ .8' 2 B.X�/ 8x 2 B.X//:

We will show that G is a B.X�/ �! G.B.X�// surjective homeomorphism.It is easy to show that G is injective. As for continuity, let f'˛g � B.X�/ be a net,

' 2 B.X�/ and '˛ ! '. Then limG.'˛/.x/ D lim.'˛; x/ D .'; x/ D G.'/.x/ for everyx 2 B.X/. It remains to prove that G�1 is continuous. Let ff˛g � G.B.X�// be a net,f 2 G.B.X�// and f˛ ! f . There exists f'˛g � B.X�/ and ' 2 B.X�/, such thatG.'˛/ D f˛ and G.'/ D f . Therefore

8x 2 B.X/ W limf˛.x/ D limG.'˛/.x/ D lim.'˛; x/ D .'; x/ D G.'/.x/ D f .x/:

40

Finally, we prove that G.B.X�// is closed in the product topology of D, it follows thatG.B.X�// is compact. Let f'˛g � B.X�/ be a net, f 2 D, and suppose that G.'˛/ ! f

in D, in other words lim.'˛; x/ D f .x/ for all x 2 B.X/. We have to show that f is abounded linear functional on X . To establish this, observe that each '˛ is linear on X , andj'˛.x/j � kxk for every x 2 X . Taking the limit shows that f 2 G.B.X�//.

(4.31) Theorem. Let X be a reflexive Banach space. Then every bounded sequence fx˛g � Xhas a weakly converging subsequence fxˇ g � fx˛g.

Proof. (From Eidelman,Milman) Let Y D spanfx˛g, then Y is closed, hence reflexive. More-over Y is separable, therefore Y � is, too, since .Y �/� D Y . Then there exists a sequencef'ˇ g � Y

�, such that Y � D spanf'ˇ g.The sequences fx˛g � X is assumed to be bounded, so f.'1; x˛/g � R is bounded. From

the Bolzano–Weierstrass theorem, we have that there exists a subsequence fx1˛g � fx˛g, suchthat �1 D lim.'1; x1˛/ exists. Then f.'2; x1˛/g � R is bounded, there is a subsequence fx2˛g �fx1˛g, such that �2 D lim.'2; x2˛/ exists. Continuing this way we obtain the subsequences

fx˛g � fx1˛g � fx

2˛g � : : : � fx

�˛g � : : :

for which.'1; x

1˛/! a1; .'2; x

2˛/! a2; : : : ; .'�; x

�˛/! a�; : : :

Consider the diagonal sequence fz˛g D fx˛˛g, a subsequence of fx�˛g˛�� for every �. Then.'ˇ ; z˛/! aˇ for every ˇ, again, because

fz�; z�C1; : : :g D fx�� ; x

�C1�C1 ; : : :g � fx

�1 ; x

�2 ; : : :g:

Now let8ˇ W ‰.'ˇ / D aˇ :

Then‰ is a real-valued functional that is linear and bounded on the dense subspace spanf'ˇ g �Y �, since taking the limit is linear, and j‰.'ˇ /j � Mk'ˇk, where M D sup kx˛k < C1.Therefore ‰ W spanf'ˇ g �! R admits a bounded linear extension b‰ to all of Y �, hence‰ 2 Y �� D Y . In other words 8' 2 Y � W .'; z˛/! .';b‰/.

To finish the proof, let ' 2 X� be arbitrary, and consider its restriction 'ˇY2 Y �. Then

.'; z˛/ D .'ˇY; z˛/! .'

ˇY;b‰/ D .';b‰/;

simply because fz˛g � Y , and b‰ 2 Y .This result is often phrased using the following terminology.

(4.32) Definition. Let X be a locally convex space. A set K � X is said to be weakly sequen-tially compact if every sequence fx˛g � K contains a subsequence fxˇ g � fx˛g and thereexists point x 2 K, such that xˇ * x.

Then the previous theorem is equivalent to the statement that

the closed unit ball of a reflexive Banach space is weakly sequentially compact.

Before stating the following fundamental theorem, we remind the reader that a subset A �X of a topological space is called limit point compact if every countable subset fx˛g � A hasa limit point x 2 A, i.e. every open punctured neighborhood of x intersects fx˛g.

41

(4.33) Eberlain–Smulian theorem. Let X be a Banach space, K � X . The following areequivalent.

(1) K is weakly sequentially compact.

(2) K is weakly limit point compact.

(3) K is weakly precompact.

(4.34) Browder’s theorem. Let X be a reflexive Banach space, A � X bounded. For every xin the weak closure of A, there exists a sequence fx˛g � A, such that x˛ * x in X .

Proof. (From De Figueiredo)

(4.35) Proposition. Let X be a reflexive Banach space, A � X bounded. A linear functional W A �! R is weakly lower semicontinuous if and only if it is weakly sequentially lowersemicontinuous.

5 DistributionsThroughout this section, � � Rn denotes an open subset.

5.1 Topology on Spaces of Continuous FunctionsFrom Treves. In this paragraph, we construct suitable topologies for the following spaces.

(1) The space C r.�/ of r times continuously differentiable, real valued functions(0 � r � 1).

(2) The space C rc .�/ of r times continuously differentiable, real-valued functionswith compact support (0 � r � 1).

C rC rC r -topology. First, we endow C r.�/ with a locally convex topology, via the family ofseminorms

8K � � compact 8s � r W jf jK;s D maxj�j�s

maxx2KjD�f .x/j:

This topologizes the locally uniform convergence (i.e. uniform on compact sets) of derivativesof order � r , that is, for a sequence ff˛g � C r.�/, f 2 C r.�/,

f˛ ! f in C r.�/ ()8K � � compact 8s � r 8j�j � s W max

x2KjD�f˛.x/ �D

�f .x/j ! 0(10)

It is easy to see that the functionals j � jK;s are seminorms, and finite, and the separation property(9) (see page 32) follows from the fact that the singletons fxg for all x 2 � are compact, henceif jf jK;s D 0, then jf .x/j D 0 for all x 2 �.

(5.1) Theorem. The space C r , 0 � r � 1, endowed with the C r -topology is a Frechet space,i.e. a complete locally convex space, metrizable with a translation-invariant metric.

42

(5.2) Lemma. Let � � Rn be open. There exists a sequence fK˛ W ˛ 2 Ng � � of compactsets such that [

˛2J

K˛ D �; and 8˛ 2 N W K˛ � intK˛C1:

Proof of the lemma. If � D Rn, then K˛ D B.0; ˛/ satisfies the required properties. If,however � ¤ Rn, let K˛ D B.0; ˛/ \ fx 2 � W d.x; @�/ � 1=˛g.Proof of the theorem. To show that the C r -topology is metrizable, we claim that it is induced bya countable subfamily of compact sets. From the preceeding lemma, we have a sequence fK˛gof compact sets such that for every K � � compact, there is an ˛ 2 N such that K � K˛, andin this case, jf jK;s � jf jK˛;s. In other words, for every K, s, and " > 0, there exists ˛, suchthat BK˛;s.0; "/ � BK;s.0; "/. Hence they two families generate the same topology.

It remains to prove that the space is complete. Let ffˇ g � C r.�/ be Cauchy seqence, thenby definition,

ffˇ g is Cauchy in C r.�/ ()8K � � compact 8s � r 8j�j � s W max

x2KjD�fˇ .x/ �D

�f .x/j ! 0 .ˇ; !1/

Let x 2 �, choose K D fxg and r D 0, we obtain that the sequence ffˇ .x/g � R is Cauchyfor every x 2 �, and by the completeness of the reals, there is a number f .x/, such thatjfˇ .x/ � f .x/j ! 0 for every x 2 �. The correspondence x 7! f .x/ is continuous function,since for every neighborhood U of x, there is a K � U compact, and on that compact set, thesequence of continuous function ffˇ g converges uniformly to f .

If r > 0, note that f@fˇ=@xkg � C r�1.�/ is Cauchy for every 1 � k � n, and convergeslocally uniformly to some function f k 2 C r�1.�/. Since we also have fˇ ! f locallyuniformly, a classical result says that f 2 C 1.�/ and in fact @f=@xk D f k. Proceeding byinduction we obtain the completeness for arbitrary r .(5.3) Remark. By the way, a possible translation-invariant metric on C r.�/ that induces theC r -topology is

d.f; g/ DX˛2Ns�r

1

2˛Csjf � gjK˛;s

1C jf � gjK˛;s:

C rcCrcCrc -topology. For every K � � compact, 0 � r � 1, consider the subspace C r.K/ �

C r.�/, defined asC r.K/ D ff 2 C r.�/ W suppf � Kg;

endowed with the normkf kK;r D max

j�j�rmaxx2KjD�f .x/j:

It is closed in C r -topology, so it inherits the Frechet space topology. Moreover it is a Banachspace if r ¤1. Then the convergence inC r.K/ is the following smooth, uniform convergence

kD�fˇ �D�f kK;r ! 0 () 8j�j � r W max

x2KjD�fˇ �D

�f j ! 0:

As in the preceeding construction, there exists a sequence fK˛g � � of compact sets, mono-tone increasing and exhausting �. Now let

C rc .�/ D[

K�� compact

C r.K/ D[˛2N

C r.K˛/;

43

where the second equality follows from the fact that for everyK � �, there exists ˛ 2 N, suchthat K � K˛, and hence kf kK;r � kf kK˛;r or, C r.K/ � C r.K˛/. Furthermore, note that

C r.K1/ � Cr.K2/ � : : : � C

r.K˛/ � : : : � Crc .�/: (11)

(5.4) Remark. The simplest way to proceed is to endow C rc .�/ with the locally convex topol-ogy generated by the family seminorms k � kK˛;s, i.e. endow it with the relative topology ofC r.�/. This space, however is not complete. Let � D R, f 2 C1c .R/ be a function whosesupport lies in Œ0; 1� and positive on .0; 1/. Then the sequence of ”bumps”

gˇ DX1� �ˇ

f .x � /

is Cauchy in the relative topology of C r . But suppgˇ D Œ0; ˇ�, hence the limit cannot possiblyhave compact support. Unfortunately, we need a more careful (and complicated) topology onC rc .�/.

For all ˛ 2 N, the relative topology on C r.K˛/, inherited from C r.�/ is the same as theone inherited from C r.K˛C1/, in other words, the inclusion map

i˛ W Cr.K˛/ �! C r.K˛C1/

is a homeomorphism. Then it makes sense to introduce the inductive limit topology on C rc .�/,defined by the seqence of Frechet spaces fC r.K˛/g. In this topology, a convex set U � C rc .�/is a neighborhood of zero in the (locally convex) inductive limit topology, if and only if forevery ˛ 2 N, the set U \ C r.K˛/ is a neighborhood of zero in C r.K˛/. It may be checkedthat this construction is independent of the choice of the sequence fK˛g, as long as it exhausts�. It can also be proved that the C rc -topology is complete, strictly coarser the relative topologygiven by C r , and induces the original topologies on the subspaces C r.K˛/, i.e. the inclusionmaps

j˛ W Cr.K˛/ �! C rc .�/

are homeomorphisms. Said differently, a set U � C rc .�/ is open if and only if j�1˛ .U / is openin C r.K˛/ for all ˛ 2 N.(5.5) Remark. The closed subspaces C r.K˛/ are proper, and it can be shown that every propersubspace of a topological vector space has empty interior. Since C rc .�/ is a countable unionof the nowhere dense sets C r.K˛/, it is not a Baire space. We conclude that C rc .�/ is notmetrizable.

Convergence in C rc .�/C rc .�/C rc .�/. Let ffˇ g � C rc .�/, f 2 Crc .�/ and fˇ ! f . Then for every

U neighborhood of zero, there is an index N , such that ˇ � N implies fˇ � f 2 U . Bydefinition, for every ˛ 2 N, U \ C r.K˛/ is a neighborhood of zero. Note that he set ffˇ g isbounded, since it is Cauchy.

(5.6) Lemma. There exists ! 2 N, such that ffˇ g � C r.K!/ and f 2 C r.K!/.

Proof. (from Rudin) Suppose for contradiction that there is no such !, i.e. fˇ ; f … C r.K˛/ forall ˛ 2 N and ˇ. Then there exists a subseqence ff g � ffˇ g, and a set fx g � � of distinctpoints without any accumulation points, such that f .x / ¤ 0. Let

V Dng 2 C rc .�/ W jg.x /j <

1

ˇjf .x /j .8 2 N/

o44

Every compact set fK˛g � � contains only finitely many of fx g, so the set V \ C r.K/ isa finite intersection of zero-centered open balls, hence it is a neighborhood of zero in C r.K/.By the definition of the inductive limit topology, V is open in C rc .�/. But f … V for any 2 N, we cannot find � > 0, such that ff g � �V , contradicting the boundedness of thesequence.

This means that the convergence in C rc .�/ is characterized as

f˛ ! f in C rc .�/ () 9K � � compact W suppf˛; suppf � K;8s � r 8j�j � s W max

x2KjD�f˛.x/ �D

�f .x/j ! 0:(12)

5.2 The Riesz–Kakutani Representation Theorem(5.7) Riesz–Kakutani representation theorem. Let X be a locally compact, separable metricspace, H a Hilbert space. Suppose that ' W Cc.X IH/ �! R is a linear functional, such thatfor every K � X compact

supf'.f / W kf k � 1; suppf � Kg < C1:

Statment. There exists

(i) a Radon measure � on X and

(ii) a �-measurable function � W X �! H , k�.x/k D 1 for a.e. x 2 H ,

and8f 2 Cc.X IH/ W '.f / D

ZX

.f; �/ d�:

Moreover for every U � X open

�.U / D supf'.f / W kf k � 1; suppf � U g:

(5.8) Riesz representation theorem for distributions. Suppose that ' is a positive linear func-tional on C1c .R

n/. Then there exists a Radon measure � on X , such that

8f 2 C1c .Rn/ W '.f / D

ZX

f d�:

Proof. First, we prove that every positive linear functional on C1c .Rn/ is bounded. LetK � Rn

be compact, and � 2 C1c .Rn/, such that 0 � � � 1 and � � 1 on K. Then for f 2 C1c .R

n/

with suppf � K, define g D kf k1� � f . Then g � 0, so by the positivity of ', we have'.f / � '.�/kf k1 for all f 2 C1c .R

n/ with suppf � K. In other words ' is bounded in thenormed space .C1c .R

n/; k � k1/.Since C1c .R

n/ is a dense subspace of the normed space Cc.Rn/, ' may be extended toa bounded (and positive) linear functional b' on Cc.Rn/. The Riesz–Kakutanai representationtheorem then yields a Radon measure � on Rn and a measurable function � W Rn �! f˙1g,such that

8f 2 Cc.Rn/ W b'.f / D Z f � d�:

Positivity ofb' however, implies � � 1 a.e. on Rn.

45

The space of Radon measures on X is denoted as M.X/. It is a Banach space when en-dowed with the total variation norm k�k D j�j.X/. Observe that if � 2M.X/, then

8f 2 C0.X/ Wˇ Z

f d�ˇ� kf k1j�j.X/ D k�kkf k1;

in other words � defines a bounded linear functional on C0.X/.

6 Sobolev spacesVarious facts about Sobolev spaces may be scattered as examples before, but we chose to putthem in their own section. The reason for this is that Sobolev spaces heavily rely on Lp spacesas primitive building blocks.

6.1 BasicsConsider the Sobolev spaces W k;p.�/ for k 2 N and 1 � p <1 defined as

W k;p.�/ D fu 2 Lp.�/ W D�u 2 Lp.�/; j�j � kg;

where� � Rn is a domain, andD� denotes weak differentiation with respect to the multiindex�, defined via

8 2 C10 .�/ W

Z�

.D�u/ dx D .�1/j�jZ�

uD� dx:

It is well-known, see Adams, that W k;p.�/ is a Banach space with the norm

8u 2 W k;p.�/ W kukW k;p.�/ D

h Xj�j�k

kD�ukp

Lp.�/

i1=p:

(6.1) Remark. It is sometimes useful to do the following trick. Let �� � Rn denote a copy of� for every j�j � k, so that f��g is disjoint, and let�k be the union of these. Now construct afunction„u 2 Lp.�k/ by copying each derivativeD�u 2 Lp.�/ to the appropriate copy��,i.e.

8j�j � k W „uˇ��D D�u:

It is easy to see that „ W W k;p.�/ �! Lp.�k/ is a linear isometry, thus the Sobolev normmay be written as an Lp-norm on a special domain: kukW k;p.�/ D k„ukLp.�k/. Of course, „is not surjective, but ran.„/ � Lp.�k/ is closed subspace. ThereforeW k;p.�/ D „�1.ran„/inherits some topological and geometric properties of the space Lp.�k/, for example we de-duce that W k;p.�/ is separable and uniformly convex (therefore reflexive, see Milman–Pettistheorem) if 1 < p <1, since Lp.�k/ is.

The dual of W k;p.�/ may be expressed by employing the aformentioned technique.(6.2) Proposition. Let 1 � p < 1. For every ' 2 W k;p.�/0 there exists v 2 Lp

0

.�k/, suchthat

8u 2 W k;p.�/ W .'; u/ DXj�j�k

.v�;D�u/;

46

where v˛ D vˇ��

and

k'k D minnkvkLp0 .�k/ W .'; u/ D

Xj�j�k

.vˇ��;D�u/ .8u 2 W k;p.�//

o:

Moreover, if 1 < p <1, then this representation is unique.

(6.3) Theorem. Let 1 � p < 1. Then W k;p.�/0 is isometrically isomorphic to the Banachspace

W �k;p0

.�/ DnT 2 D.�/0 W 9v 2 Lp

0

.�k/ W T DXj�j�k

.�1/j�jD�Tv�

o;

endowed with the norm

kT kW �k;p0 .�/ D minnkvkLp0 .�k W T D

Xj�j�k

.�1/j�jD�Tv�

o:

6.2 RegularizationRegularization, a.k.a mollification is the convolution ofLp.Rn/ function f by a suitableC1.Rn/smoothing kernel k supported onB.0; 1/, so that forR"f D k"�f , we have kR"f �f kp ! 0

(" ! 0). The most important fact is that R"f 2 C1.Rn/, and R" W Lp.Rn/ �! L.Rn/ is abounded linear operator. The fact that C1.Rn/ is dense in Lp.Rn/ is also follows immediately.

(6.4) Theorem. Let k" 2 C1.Rn/ be a smoothing kernel.

(1) If f 2 L1loc.Rn/, then for very " > 0, R"f 2 C1.Rn/ and for every multiin-

dex �, we have D�.k" � f / D .D�k"/ � f .

(2) If f 2 L1loc.Rn/, then R"f ! f a.e. on Rn as " & 0, and if f continuous,

then R"f ! f uniformly on compact sets.

(3) R" W Lp.Rn/ �! Lp.Rn/ is a bounded linear operator with kR"k � 1, andkR"f � f kp ! 0 as "& 0.

Proof. To prove (2),

j.R"f � f /.x/j Dˇ Z

k".x � y/f .y/ dy � f .x/ˇ�

ZB.x;"/

k".x � y/jf .y/ � f .x/j dy

D1

"n

ZB.x;"/

k.x � y/jf .y/ � f .x/j dy � kkk11

"n

ZB.x;"/

jf .y/ � f .x/j dy;

where the last integral average tends to 0 for a.e. x 2 Rn, due to Lebesgue’s differentiationtheorem. The set on which the convergence takes place is called the set of Lebesgue points off . The statement on the locally uniform convergence is because f is uniformly continuouson compact sets, and we may appeal to the familiar integral convergence theorem of Riemannintegral.

Finally for (3), since R" is really just a convolution, we may use Young’s inequality withp D r and q D 1, so

kR"f kp D kk" � f kp � kk"k1kf kp;D kf kp

for f 2 Lp.Rn/, hence kR"k � 1. Moreover, using the boundedness of R" and the density ofC10 .R

n/ in Lp.Rn/ we may deduce that kR"f � f kp ! 0 for every f 2 Lp.Rn/.

47

6.3 EmbeddingsWe begin by calculating the action of a dilation in W k;p.Rn/. Since Lp.Rn/ is dilation-invariant, W k;p.Rn/ is, too, but the norms change in a different manner. More precisely, let

.ı�u/.x/ D u.�x/

denote the dilation of a function u W Rn �! R by the factor � > 0. If u 2 Lp.Rn/, then

kı�ukpp D

Zju.�x/jp dx D ��n

Zju.y/jp dy D ��nkukpp ;

and if u 2 W k;p, j�j � k, then

kD�ı�ukpp D

Z ˇD��u ı .� Id/

�.x/ˇpdx D �j�jp

Zj.D�u/.�x/jp dx

D �j�jp�nkD�ukpp :

In this section, we first investigate the validity of bounded embeddings of the form

W k;p.Rn/ � Lq.Rn/ (13)

Let’s start with the case k D 1, for k D 0 admits the only solution q D p. If we somehowobtain result for the case k D 1, i.e. we manage to find set of values of q for which (13) holds,we assure the reader that the general case will follow by iteration. Let us suppose that for k D 1we managed to find the inequality

kukq � Cn;p;qkDukp;

for functions u 2 C 1c .Rn/. If u is such a function, then obviously ı�u 2 C 1c .R

n/ for every� > 0 so the hypothetical inequality holds for ı�u too, and the norms of dilated functions canbe expressed with the norms of u of Du,

��nq kukq D kı�ukq � Cn;p;qkDı�ukp D �

1�np kDukp;

orkukq � �

1�npCnqCn;p;qkDukp:

In other words, the constant Cn;p;q actually contains that exponential factor for all � > 0! Butthat is only possible if

1 �n

pCn

qD 0;

because either �& 0 or �%C1 would lead to a contradiction. Hence the exponent 1 � q <1 is constrainted to be p�, called the Sobolev conjugate of p,

1

p�D1

p�1

n; or, equivalently, p� D

n

n � p;

and we have the additional constraint p < n, since otherwise no such 1 � q < 1 couldbe found. This technique is called a homogenity argument, and is very useful for conjectureor disproof of various inequalities. Of course, it does not prove anything – it merely imposesconstraints (which might not be sufficient) under which the statement is conceivable at all.

48

(6.5) Gagliardo–Nirenberg–Sobolev inequality. Suppose that 1 � p < n. Then there existsa constant Cn;p > 0, such that

8u 2 C 1c .Rn/ W kukp� � Cn;pkDukp:

Proof.Note that iterating the ”Sobolev conjugation” k times on an exponent 1 � p, such that

k < np

, where k � 0 an integer

1

p�D1

p�1

n

1

p��D

1

p��1

nD1

p�2

n:::

1

p�:::�D1

p�k

n;

which is still positive by assumption.

6.4 Riesz PotentialsLet n > a > 0, f 2 S.Rn/ and define the Riesz potential (or Riesz transform) of f with theinstruction

8x 2 Rn W .Iaf /.x/ D Cn;a

Zkx � yk�nCaf .y/ dy;

where Cn;a is some normalization constant. There are different ways to motivate this definition.First, Iaf D k � k�nCa � f . From Example (2.3) (page 10), we have that k � k�nCa 2 Lp

loc.Rn/

for all 1 � p < 1, but k � k�nCa … Lp.Rn/. Convolution with functions with singularitiesmight be familiar from potential theory.

A different way of looking at things is through the glass of Fourier transform. Note thatk � k�nCa … L1.Rn/, so it does not have a Fourier transform in the L1 sense, but it is locallyintegrable, hence it generates a regular distribution.

(6.6) Proposition. Let 0 < a < n, then for all f 2 S.Rn/,Zkxk�nCaf .x/ dx D

1

Cn;a

Zk�k�abf .�/ d�:

Note that the right-hand integral is the precisely the regular distribution generated by k � k�nCa

evaluated at f . Since Fourier transform maps S.Rn/ to itself, we have

.3k � k�nCa; f / D .k � k�nCa;bf / D Cn;a.k � k�a; f /;in other words the Fourier transform of k � k�nCa is proportional to k � k�a.(6.7) Remark. Before turning to the proof, note that from the definition of the gamma function,

�.b/ D

Z C10

tb�1e�t dt;

49

we deduce by a change of variables we that for every c > 0,Z C10

tbe�tc dt D c�b�1Z C10

sbe�s ds D c�b�1�.b C 1/:

(6.8) Remark. The Schwartz function W.x/ D e�kxk2

2 is an eigenvector of the Fourier trans-form for the eigenvalue 1. First for n D 1, let

F.�/ D bW .�/ D Z C1�1

e�x2

2 e�ix� dx:

Then

F 0.�/ D

Z C1�1

.�ix/e�x2

2 e�ix� dx D i

Z C1�1

W 0.x/e�ix� dx

D ��

Z C1�1

W.x/e�ix� dx D ��F.�/:

This relation, F 0.�/ D ��F.�/, together with the initial condition F.0/ D 1 is an initial valueproblem that admits a unique solution. Now observe that W is a solution, therefore the onlysolution, which means that bW D W . The case n > 1 follows easily, since

e�k�k2

2 D

Y1�˛�n

e��2˛2 ; and e�i.x;�/ D

Y1�˛�n

e�i.x˛;�˛/:

Therefore

bW .�/ D Z C1�1

� � �

Z C1�1

Y1�˛�n

e��2˛2 e�i.x˛;�˛/ dx1 : : : dxn

D

Y1�˛�n

Z C1�1

e��2˛2 e�i.x˛;�˛/ dx˛ D

Y1�˛�n

e��2˛2 D W.�/:

Proof of the proposition. From the dilation property of the Fourier transform, we have for allt > 0 that

7�x 7! e

�tkxk2

2

�.�/ D3.ıt1=2W /.�/ D t�n=2bW �

t�1=2��D t�n=2W

�t�1=2�

�D t�n=2e

�k�k2

2t :

Hence using the remark on the Gamma function with b D a2� 1, and c D kxk2=2Z

kxk�af .x/ dx D1

2a2

Z �kxk2

2

��a2f .x/ dx D

1

2a2��a2

� Z Z C10

ta�22 e�tkxk2

2 f .x/ dt dx

D1

2a2��a2

� Z C10

ta�22

Ze�tkxk2

2 f .x/ dx dt:

50

Using Parseval’s identity .f; g/L2 D .2�/�n.bf ;bg/L2 ,1

2a2 .2�/n�

�a2

� Z C10

ta�22

Zt�n=2e

�k�k2

2t bf .�/ d� dtD

1

2a2 .2�/n�

�a2

� Z bf .�/ Z C10

ta�n2�1e�k�k2

2t dt d�:

The inner integral is nothing butZ C10

ta�n2�1e�k�k2

2t dt D k�ka�nZ C10

sa�n2�1e�12s ds

D k�ka�n1

2a�n2�1

Z C10

un�a2C1e�u

u�2

2du

D k�ka�n��n�a2

�2a�n2

using the substitutions t D k�k2s, dt D k�k2ds and s D .2u/�1, ds D �2.2u/�2du. Theconstant Cn;a is then probably something like

1

Cn;aD

��n�a2

�2aCn�n�

�a2

�An easy consequence of the proposition is thatZ

.Iaf /.x/g.x/ dx D

Z bf .�/k�k�abg.�/ d�;because multiplying the equation

.Iaf /.x/ D Cn;a

Zkyk�nCaf .x � y/ dy D

Zk�k�abf .�/e�i.x;�/ d�;

by g.x/ and integrating, yields the desired relation. This result is interpreted as

1.Iaf /.�/ D k�k�abf .�/;to be understood in sense of distributions.

Our main reason for studying Riesz potentials is to establish estimates of the form

kIaf kq � Cn;a;pkf kp: (14)

First, we derive some restrictions on the exponent q. Suppose that (14) holds, and take � > 0,.ı�f /.x/ D f .�x/. Then

.Iaı�f /.x/ D Cn;a

Zkx � yk�nCaf .�y/ dy D Cn;a�

�n

Zkx � ��1zk�nCaf .z/ dz

D Cn;a��a

Zk�x � zk�nCaf .z/ dz D ��a.ı�Iaf /.x/

51

Then kIaı�f kq D ��a�

nq kIaf kq and kı�f kp D �

�np kf kp, hence

kIaı�f kq � �aC

nq�np kf kp;

then necessarily1

qD1

p�a

n; and a <

n

p;

in other words q D p� the Sobolev exponent.

7 Linear Elliptic Boundary Value Problems

7.1 The Courant–Fischer Principle

8 Critical Point Theory

8.1 Variational Principles and Compactness(8.1) Ekeland’s variational principle. Let .X; d/ be a complete metric space, and let j WX �! R [ fC1g be

(A) lower semicontinuous,

(B) bounded from below.

Suppose that " > 0, � > 0 and x 2 X such that

j.x/ � infXj � ":

Statement. There exists y 2 X , for which

(1) j.y/ � j.x/

(2) d.x; y/ � 1=�

(3) 8z 2 X X fyg W j.y/ � j.z/ < "�d.y; z/:

Proof. (From Chabrowski) It is enough to assume that � D 1, since in general the metric �d isequivalent to d and yields the result.

Introduce a partial order � on X via

8z; w 2 X W z � w () "d.w; z/ � j.w/ � j.z/:

It is obviously reflexive, antisymmetry also follows easily. As for transitivity, suppose thatz � t and t � w, i.e.

"d.t; z/ � j.t/ � j.z/

"d.w; t/ � j.w/ � j.t/:

Then"d.w; z/ � "d.t; z/C "d.w; t/ � j.w/ � j.z/:

52

Construct a sequence of points fx˛g � X and of sets fS˛g � P .X/, by letting x0 D x andinductively, supposing x˛ is already defined, let

S˛ D fw 2 X W w � x˛g;

andx˛C1 2 S˛; j.x˛C1/ � inf

S˛j �

1

˛ C 1:

Some explanation is in order. An element y 2 X is minimal with respect to �, if for all z 2 X ,such that z � y we have y D z, which is equivalent to y ¤ z implies z 6� y, i.e.

8z 2 X W y ¤ z H) "d.y; z/ > j.y/ � j.z/;

which is precisely (3). Furthermore, x˛C1 is chosen so as to be closer and closer to the infimumthat is achievable on the set S˛ – and simultaneously marching toward minimality.

In fact, we have x˛C1 � x˛, due to S˛C1 � S˛. Moreover S˛ is actually a level set of j ,

S˛ D fw 2 X W j.w/ � j.x˛/ � "d.x˛; w/g D fw 2 X W j.w/ > j.x˛/ � "d.x˛; w/gc;

therefore S˛ is closed, since j is lower semicontinuous. To apply Cantor’s intersection theorem,we have to show that all S˛ is nonempty and diamS˛ ! 0. First suppose on the contrary thatthere exists ˇ, such that Sˇ D ;. Then xˇ � z for all z 2 X , i.e.

8z 2 X W "d.z; x˛/C j.x˛/ � j.z/;

in particular, we get that j is not bounded form below, a contradiction. As for diamS˛ ! 0,let w 2 S˛C1, then w � x˛C1 � x˛, by definition. The first relation and the defintion of x˛C1implies

"d.x˛C1; w/ � j.x˛C1/ � j.w/ � infS˛j C

1

˛ C 1� j.w/

� infS˛j C

1

˛ C 1� infS˛j D

1

˛ C 1;

from which the claim follows. Cantor’s intersection theorem implies the existence of y 2 X ,such that \

˛�0

S˛ D fyg:

This y is a minimal, since if z � y, then z � x˛ for all ˛, in other words z 2TS˛, hence

z D y.It remains to show (1) and (2), the first one follows from y 2 S0, so y � x0 D x, thus

j.y/ � j.y/C "d.x; y/ � j.x/;

and the second one from

d.x; y/ �1

"

�j.x/ � j.y/

��1

"

�infXj C " � inf

Xj�D 1

53

(8.2) Remark. Consider the function

8z 2 X X fyg W �.z/ D j.y/ � "�d.y; z/:

Then (3) of Ekeland’s principle states that�.z/ < j.z/. Note that the graph of� is an upward-pointing cone (without its vertex) in X � R, entirely contained in the subgraph of j . The”perturbation term” z 7! �"�d.y; z/ is obviously Lipschitz continuous. But if differentiationmakes sense, i.e. X is a normed space, then the perturbation is z 7! �"�ky � zk, which is notdifferentiable at y in general. To remedy this, a number of smooth variational principles weredeveloped, see later in this section.

(8.3) Corollary. Let X be a Banach space, j 2 C 1.X/ bounded from below. Then there existsa minimizing sequence fu˛g � X , such that j.u˛/! infX j and Dj.u˛/! 0 in X�.

Proof. Let "˛ & 0, �˛ D 1=p"˛. Then for every ˛ 2 N, let

j.u˛/ � infXj � "˛:

By Ekeland’s variational principle, there exists v˛ 2 X , such that j.v˛/ � j.u˛/, d.v˛; u˛/ �p"˛ and

8w 2 X X fv˛g Wj.v˛/ � j.w/

kv˛ � wk<p"˛;

or, by letting w D v˛ C h,

8h 2 X X f0g Wj.v˛/ � j.v˛ C h/

khk<p"˛:

Then

kDj.v˛/k D supkhk�1h¤0

j.Dj.v˛/; h/j D supkhk�1h¤0

limkhk!0

jj.v˛ C h/ � j.v˛/j

khk�p"˛

The following compactness criterion imposed on the functional j assures that if a sequenceis bounded in energy, and it is eventually critical, then a subsequence can be selected from itthat is actually converging to a minimizer.

(8.4) Definition. Let X be a Banach space. The functional j 2 C 1.X/ is said to satisfy thePalais–Smale compactness condition (PS) if for every fu˛g � X sequence, such that fj.u˛/g �R is bounded and Dj.u˛/! 0 in X�, there exists a convergent subsequence fuˇ g � fu˛g.

Using this terminology, we have the following convenient formulation.

(8.5) Corollary. Let X be a Banach space, j 2 C 1.X/ bounded from below and supposethat j is a Palais–Smale functional. Then there exists a minimizing critical point u 2 X , i.e.j.u/ D infX j and Dj.u/ D 0.

In fact it suffices to assume that the Palais–Smale compactness condition applies only at thelevel set fj D infX j g D fu 2 X W j.u/ D infX j g.

(8.6) Definition. Let X be a Banach space. The functional j 2 C 1.X/ is said to satisfy thePalais–Smale compactness condition at level c 2 R (PSc) if for every fu˛g � X sequence, suchthat j.u˛/! c and Dj.u˛/! 0 in X�, there exists a convergent subsequence fuˇ g � fu˛g.

54

8.2 ???(8.7) Proposition. Let K � X be convex. Then K is closed in X iff it is closed in X� .

(8.8) Definition. Let X be a Banach space, a subset Y � X is said to be weakly sequentiallycompact, if from an arbitrary sequence fx˛g � Y , one can select a subsequence fxˇ g � fx˛gand a point x 2 X , such that xˇ * x.

Suppose fx˛g � Y is a minimizing sequnce of some functional f W Y �! R [ fC1g,i.e. f .x˛/& infY f , such sequence exists by the definition of infimum. Now select a weaklyconvergent subsequence fxˇ g � fx˛g and a point x, such that xˇ * x.

We may now relax the continuity of f assumed in Weierstrass’ theorem.

(8.9) Definition. The functional f W Y �! R[fC1g is said to be weakly sequentially lowersemicontinuous in Y , if

8fy˛g � Y 8y 2 Y W y˛ * y H) f .y/ � lim inff .y˛/:

This way, we can write

infYf � f .x/ � lim inff .xˇ / D limf .xˇ / D inf

Yf;

so infY f D f .x/. Similar deductions are true for the supremum if the hypotheses are properlyaltered.(8.1) Example. Can we perhaps furnish some examples to the concepts introduced? Certainly.It is well-known (See Adams, Theorem 1.18) that the fact the unit ball is weakly sequentiallycompact characterizes reflexive Banach spaces. As for weak sequential lower semicontinuity,remember that a weakly convergent sequence fx˛g � Lp.�/, x˛ * x, is bounded in Lp.�/and kf kp � lim inf kf˛kp. This means that the Lp-norm posess this property. Investigation ofweak sequential lower semicontinuity of functionals of the form

u 7!

Z�

L.x; u;Du/ dx

is of paramount importance for the calculus of variations to work. A number of interestingexamples will be presented later on.(8.10) Remark. There is a more general definition of lower semicontinuity applicable whereno sequences are available. A funcion f W X �! R [ fC1g is called lower semicontinuous,if for every a 2 R, the level set ff > ag is open inX . It involves a little knowledge of topologyto show that for metric spaces the two definitions coincide.

As a reminder and for later reference, we summarize this paragraph with

(8.11) Theorem. Let X be a Banach space, Y � X weakly sequentially compact, and f WY �! R [ fC1g weakly sequentially lower semicontinuous. Then f attains its infimum inY .

Returning to the general compact space X – note that sequential compactness may be dif-ferent now –, let us look at the level sets of a lower semicontinuous function f W X �!R [ fC1g. Let

8˛ 2 N W U˛ ´ ff > �˛g;

55

then fU˛ W ˛ 2 Ng is an open covering of X . Using the compactness of X , there exists afinite index set J � N, such that fU˛ W ˛ 2 J g also covers X . In other words

f > �minJ;

that is, f is bounded from below. To show that the infimum of f is attained in X , we shallreach a contradiction by assuming infX f is not attained in X . That is to say

X D[˛2N

ff > 1˛C inf

Xf g;

but by compactness, there exists ˇ 2 N, such that

X D[˛�ˇ

ff > 1˛C inf

Xf g D ff > 1

ˇC inf

Xf g;

where last equality follows from the fact that the level sets are monotone increasing in ˇ. Thiscontradicts the very definition of infX f , since we have

9ˇ 2 N W f � infXf > 1

ˇ.on X/:

In summary we have

(8.12) Theorem. Let X be a compact topological space and f W X �! R [ fC1g lowersemicontinuous. Then f is bounded from below and infX f is attained.

These innocent-looking theorems are the main pillars of critical point theory.

9 Nonlinear Problems

10 Semicontinuity and Quasiconvexity

10.1 Nemytskii OperatorsLet us consider substitution operators, i.e. operators of the form G.f /.x/ D g.x; f .x//, thatbehave reasonably in a functional analytic sense. But first of all a very important technicaldefinition.

(10.1) Definition. A function g W � � R �! R, where � � Rn is open, is said to beCaratheodory, if

(1) 8� 2 R W g. � ; �/ measurable,

(2) 8x 2 � W g.x; � / continuous.

It is obvious that functions in C.� � R/ are Caratheodory. Next we prove the unsurprisingfact that a substitution into a Charatheodory function does not ruin measurability.(10.2) Lemma. Let g be a Caratheodory function. Then if f W � �! R is measurable, thenG.f /.x/ D g.x; f .x// is measurable.

56

Proof. Let f be a step function, with J a finite index set, fc˛g � R, fE˛g � � measurable and

f DX˛2J

c˛�E˛ :

Now f is piecewise constant, therefore substitution is particularly simple,

g.x; f / DX˛2J

g.x; c˛/�E˛.x/ .a.e. on �/;

which is measurable since g is measurable in its first argument, �E˛ is measurable, and productsand linear combinations are measurable.

Next, construct a sequence ff˛g of step functions converging a.e. to an arbitrary measurablefunction f . Then, since g is continuous in its second argument

g.x; f˛.x//! g.x; f .x// .a.e. on �/;

which is measurable, since it is a limit of measurable functions.

(10.3) Theorem. Let g W � � R �! R be a Caratheodory function. Suppose furthermore thatthere exists b; r > 0, a 2 Lp.�/ for some 1 � p � 1, such that

a.a. x 2 � 8� 2 R W jg.x; �/j � a.x/C bj�jr : (15)

Then the Nemytskii operator generated by g is a continuous map

G W Lpr.�/ �! Lp.�/;

and it maps bounded sets to bounded sets if j�j <1 or a D 0.

Proof. (From De Figueiredo) An application of Minkowski’s inequality yields for all u 2Lpr.�/,

kG.u/kp D kg. � ; u/kp � kakp C bkjujrkp D kakp C bkuk

rpr :

As for continuity, suppose that u˛ ! u in Lpr.�/. For every subsequence fuˇ g � fu˛g, byRiesz’s selection lemma (2.9) (page 10), there exists a subsequence fu g � fuˇ g such thatu ! u a.e. and ju j � m a.e. on � for some m 2 Lpr.�/. By the continuity of g in thesecond argument, G.u /! G.u/ a.e. on �, and

jG.u /j � jaj C bju jr� jaj C bmr 2 Lp.�/:

Hence, by Lebesgue’s theorem we have kG.u /�G.u/kp ! 0. By Cantor’s lemma, a sequencein a topological space converges to a limit if (and only if) every subsequence has a subsequenceconverging to that limit.

(10.4) Theorem. Assume g is a Caratheodory function, and that the corresponding Nemytskiioperator G.u/ D g. � ; u/ is a map

G W Lp.�/ �! Lq.�/;

for some 1 � p; q <1.Statement. G is continuous, bounded and there exists b; r > 0, a 2 Lq.�/, such that

a.a. x 2 � 8� 2 R W jg.x; �/j � a.x/C bj�jp=q:

57

(10.5) Remark. A Nemytskii operator is only compact Lp �! Lq in the uninteresting casewhen g.x; �/ D a.x/ for some a 2 Lq.

Next, we turn to the question of differentiability, and the computation of the derivative of aNemytskii operator.

(10.6) Theorem. Assume that g is a Caratheodory function, and that the corresponding Nemyt-skii operator G.u/ D g. � ; u/ is a map

G W Lp.�/ �! Lq.�/;

wheres > 0; 1 � t � 1; p D st; q D

st

s C 1:

Furthermore, assume that @g.x;�/@�

is also Caratheodory, and that there exists a 2 Lt.�/ andb > 0, such that

a.a. x 2 � 8� 2 R Wˇ@g.x; �/

@�

ˇ� a.x/C bj�js: (16)

Statement. We have G 2 C 1.Lp.�/; Lq.�// and

DG W Lp.�/ �! B.Lp.�/; Lq.�//; DG.u/v D@g. � ; u/

@�v .8u; v 2 Lp.�//:

Proof. First, we confirm that the growth condition (16) implies the growth condition (15). Inte-grating (16) with respect to � ,

jg.x; �/j � j�ja.x/Cb

s C 1j�jsC1 C C.x/

�1

s C 1j�jsC1 C

s

s C 1a.x/.sC1/=s C

b

s C 1j�jsC1 C C.x/

Db C 1

s C 1j�jsC1 C

s

s C 1a.x/.sC1/=s C C.x/:

Then a.sC1/=s 2 Lq.�/, and by choosing C 2 Lq.�/, we get the desired properties.Next, we check that @g. � ;u/

@�v 2 Lq.�/. Indeed, by the generalized Holder’s inequality @g. � ; u/@�

v q�

@g. � ; u/@�

tkvkp;

after a little juggling with the exponents. But @g. � ;�/@�

itself induces a Nemytskii operator H W

Lp.�/ �! Lt.�/, where p D st .To prove thatG is Frechet differentiable and its derivative is the specified function, we have

to check that the function

A.v/ D G.uC v/ �G.u/ �@g. � ; u/

@�v

satisifies kA.v/kq=kvkp ! 0 as kvkp ! 0. By the Newton–Leibniz formula

g. � ; uC v/ � g. � ; u/ D

Z 1

0

d

d�g. � ; uC �v/ dt D

Z 1

0

@g. � ; uC �v/

@�v d�;

58

then by using Minkowski’s integral inequality and Holder’s inequality

kA.v/kq D Z 1

0

�@g. � ; uC �v/@�

�@g. � ; u/

@�

�v d�

q

�

Z 1

0

�@g. � ; uC �v/@�

�@g. � ; u/

@�

�v qd�

�

Z 1

0

@g. � ; uC �v/@�

�@g. � ; u/

@�

td� kvkp

D

Z 1

0

kH.uC �v/ �H.u/kt d� kvkp

from which the required convergence follows, because H is continuous.(10.7) Remark. Consider the case s D 0, t D 1, i.e. @g.x;�/

@�is bounded on � � R. If

G W Lp.�/ �! Lp.�/ (1 � p � 1) is Frechet differentiable, then there exists a 2 L1.�/,b 2 Lp.�/, such that g.x; �/ D a.x/� C b.x/. In other words, G is necessarily linear.

We finally turn to the computation of the potential of a Nemytskii operator, the mostcommon form used in calculus of variations. Define the functional ˆ W Lp.�/ �! R(1 � p � 1), with

ˆ.u/ D

Z�

�. � ; u/ dx; �. � ; �/ D

Z �

0

g. � ; �/ d�:

The above definition makes sense if and only if �. � ; u/ 2 L1.�/ for u 2 Lp.�/. To this end,we suppose that there exists s > 0, c > 0 and a 2 Lp=s.�/, such that

a.a. x 2 � 8� 2 R W jg.x; �/j � a.x/C bj�js;

and from Theorem (10.3), we have that G W Lp.�/ �! Lp=s.�/ is continuous. Using thenotation �. � ; �/ for the primitive, we have

a.a. x 2 � 8� 2 R W j�.x; �/j � A.x/C b0j�jsC1;

where A 2 Lp=.sC1/.�/. Again, from Theorem (10.3), we have that the Nemytskii operatorcorresponding to � , denoted as Z W Lp.�/ �! Lp=.sC1/.�/ is continuous. The choicep D s C 1 leads to the desired conclusion, in that case G W Lp.�/ �! Lp

0

.�/ and Z WLp.�/ �! L1.�/ are continuous, where p0 D p=.p � 1/ D .s C 1/=s.

(10.8) Theorem. Assume that g is a Caratheodory function. Suppose that 1 < p � 1, p0 Dp=.p � 1/ and there exists constants b > 0, function a 2 Lp

0

.�/, such that

a.a. x 2 � 8� 2 R W jg.x; �/j � a.x/C bj�jp�1:

Then ˆ 2 C 1.Lp.�//, i.e. ˆ is continuously Frechet differentiable. Moreover, Dˆ D G.

Proof. In view of the above considerations it suffices to show that Dˆ D G, in other words, ˆis a potential of the Nemytskii operator generated by g. Consider the function

8v 2 Lp.�/ W B.v/ D ˆ.uC v/ �ˆ.u/ � .G.u/; v/

D

Z�

�. � ; uC v/ � �. � ; u/ � g. � ; u/v dx

59

so we have to prove that jB.v/j=kvkp ! 0 as kvkp ! 0. Similarly to the proof of Theorem(10.6), write

�. � ; uC v/ � �. � ; u/ D

Z 1

0

d

d��. � ; uC �v/ d� D

Z 1

0

d

d�

Z uC�v

0

g. � ; �/ d� d�

D

Z 1

0

g. � ; uC �v/v d�:

Therefore

jB.v/j � Z 1

0

�g. � ; uC �v/ � g. � ; u/

�v d�

1�

Z 1

0

�g. � ; uC �v/ � g. � ; u/�v 1d�

�

Z 1

0

g. � ; uC �v/ � g. � ; u/ p0d� kvkp D

Z 1

0

kG.uC �v/ �G.u/kp0 d� kvkp:

The desired conclusion then follows from the continuity of G.

11 Numerical Methods

11.1 The Newton–Kantorovich Theory

11.2 Multilevel Methods

60