linear functionals satisfying prescribed conditions

LINEAR FUNCTIONALS SATISFYING PRESCRIBED CONDITIONS

RALPH PALMER AGNEW

1. Introduction. A function (or transformation) q =- q(x) with domain andrange in linear spaces is called linear if

(1.01) q(ax - by) aq(x) - bq(y) (a, b R; x, y F_,),

where E is the domain of q and R is the set of real numbers. If the range ofq(x) is in R, then q(x) is called a functional. Using notation of Banach we call afunctional p(x) a p-function if

(1.02) p(tx) tp(x) (t >- 0; x E),

(1.03) p(x d- Y) <- p(x) - p(y) (x, y E).

We denote the class of linear functionalsf =- f(x) by F and the class of p-functionsby P.A theorem of Banach (loc. cit., p. 29) of which we make repeated use isTHEOREM 1.1. If p e P, then there exists f F with

f(x) -< p(x)

Since each linear functional f is also a p-function, i.e., F C P, the followingtheorem, of which we shall make explicit and implicit use, is trivial.THEOREM 1.2. If f e F, then there exists p P with f(x) <= p(x) for all x E.Let p0 e P and a set of pairs {x, y} of elements x, y e E be prescribed. One

problem in which we shall be interested is that of determining whether thereexist linear functionals f e F possessing the properties

(1.21) f(x) <= po(x)

(1.22) f(y) f(z) ({ x, y} 9).

We assume xI, has the property that if {x, y} e xI, then {y, x} e , and that{x, x} e xI, for each x e E; this assumption is convenient and entails no loss ofgenerality.We shall say that a p-function p p(x) enforces a specified property (or set of

properties) if everyf e F, with f(x) <= p(x) for all x e E, must possess the specifiedproperty (or set of properties).For example, a slight amplification of work of Banach (loc. cit., p. 33) shows

Received September 27, 1937; presented to the American Mathematical Society, October30, 1937.

S. Banach, Thorie des Operations Lin$aires, Warsaw, 1932, p. 28.

55

56 RALPH PALMER AGNEW

that if E is the space of real bounded functions x x(s) defined overs < , then

(1.23) pB(x) g.l.b, lim 1_ x(s -t-n>0; k R --* n kl

is a p-function which enforces the properties

(1.24) f(x) <- lim x(s) (x E),

and

(1.25) f(x(s - ))) f(x(s)) (he R; x e E).

The interest in f(x) lies in the fact that Lim x(s) ---f(x) is a generalization of

lim x(s) which exists for all real bounded functions. The rSle of the analogue

(1.24) of (1.21) is to ensure that the generalized limit of x(s) lies between theinferior and superior limits of x(s); and the rSle of the analogue (1.25) of (1.22)is to ensure that the generalized limits of x(s) and x(s - )) are equal. Thisexample and related ones will be discussed in 9.There is of course no priori reason for believing that there exists p e P

which enforces specified properties. The situation is governed byTHEOREM 1.3. In order that there exist f F having a specified set of properties,

it is necessary and sucient that there exist at least one p P which enforces theseproperties.

Proof of this theorem is quite trivial. To prove SUfficiency, choose pl e Pwhich enforces the properties. By Banach’s Theorem 1.1, there exists fl Fwith fl(x) <= p(x) and hence this f must have the properties. To prove neces-sity, let f F have the properties. Thenf itself is a p-function which enforcesthe properties;for if f(x) is a linear functional with f(x) <= fl(x) for all x e E, then--f(x) f(-- x) <= f(-- x) --f(x), so f(x) <= f(x) for all x E. Hencef(x)f(x), and f(x) has the properties in question. This proves the theorem.

The preceding definitions and theorems suggest the main problem of thispaper, namely, that of characterizing analytically the class of p-functions whichenforce a specified property or set of properties. The part of the paper from 4onward deals largely with problems of this type. 2 and 3 give lemmasinvolving p-functions and r-functions needed in later sections.

It is known that, if G is a solvable group as in 8 and p0 P is such thatpo(g(x)) pox for all g e G, x e E, there exists a linear functional f with theproperties

(1.41) f(x) <- po(x) (x E),(1.42) f(g(x)) f(x) (g e G; x e E).

R. P. Agnew and A. P. Morse, Extensions of linear functionals, with applications tolimits, integrals, measures, and densities, Theorem 3. This paper is to appear in the Annalsof Mathematics.

LINEAR FUNCTIONALS AND PRESCRIBED CONDITIONS 57

Here I, is the set of all pairs {x, g(x) obtained by taking g e G, x E. In 8 wegive a specific p-function which enforces (1.41) and (1.42), and characterizesthe class of p-functions which enforce these properties. 9 and 10 give appli-cations to limits and integrals.

2. Properties of p-functions. In this section, we give as lemmas some proper-ties of p-functions of which explicit and implicit use will be made later.LEMMA 2.1. In order that a functional p(x) may be a p-function, it is necessary

and sucient that

(2.11) p(tx) tp(x), p(x + y) <-_ p(x) + p(y) (t > 0; x, y E).

Necessity is obvious from (1.02) and (1.03). To prove sufficiency, we require,in addition to (2.11), only the property

(2.12) p(0) 0;

and this follows on putting x 0 and 2 in the first formula of (2.11).LEMMA 2.2. If p P and Xo E, then a necessary and sucient condition that

(2.21) p(x + Xo) p(x) (x E)

is that p(+/-xo) O, i.e., p(Wxo) 0 and p(-xo) O.To prove necessity, suppose (2.21) holds. Using (2.11), (2.12) and the

results obtained by setting x -x0 and x x0 in (2.21), we find p(:t:x0) 0.To prove sufficiency, suppose p(xo) 0. Then use of (2.11) gives

p(x - Xo) <= p(x) + p(xo) p(x),

and

p(x) p[(x - Xo) -x0] _-< p(x + Xo) + p(-xo) p(x + Xo),

from which (2.21) follows.LEMMA 2.3. If p P, then the set Eo of x E for which p(:i=x) 0 forms a

linear manifold in E.This is easily proved with the aid of Lemma 2.2. The set of x e E for which

p(x) 0 does not ordinarily form a linear manifold in E.LEMMA 2.4. If p P; xl x E; and p[(x- xl)] O, then p(x) p(x).This follows from Lemma 2.2 since it justifies writing

p(x) p[xl +(x.- x)] p(x.).

LEMMA 2.41. /f p e P; xl, x. e E; and p(x.) p(xl), then p[-4- (x. Xl)] _- 0.From p(x.) p[(x.- Xl) -xl] =< p(x- x) p(x) and our hypothesis

follows p(x xl) >= O. Likewise p(x x) >= 0 and Lemma 2.41 is proved.LEMMA 2.5. If p P, then

(2.51) p(-- x) <= p(x) and p(x) <- p(- x) (x E).

58 RALPH PALMER AGNEV

The conclusions in (2.51) are obtained by transpositions in the inequality

(2.52) 0 p(O) p(x- x) <- p(x) -4- p(-x) (xe E).

It is easy to show that the reverse inequality p(x) <-_ -p(-x) cannot hold forall x e E unless p(x) is linear.

If p P and, for some Xo E, p(-+-xo) <-_ O, then p( +xo) O.If pl p+. P and

LEMMA 2.6.LEMMA 2.7.

(2.71)then

(2.72)

LEMMA 2.8.

(2.81)

LEMMA 2.9.

p,() __< p() (x E),

p.(- x) <-_ p(- x) <- p(x) <-_ p,.(x) (x E).

Iff F, p P, and f(x) <-_ p(x) for all x E, then

p(- x) <-_ f(x) <- p(x) (x E).

If p P and . is a linear transformation with domain and rangein E, then the functional defined by

(2.91) p(x) p(’(x))

is a p-function.Since F C P, Lemma 2.8 is a corollary of Lemma 2.7.

2.6, 2.7, and 2.9 are left to the reader.Proofs of Lemmas

3. Properties of r-functions. We now give, for future reference, the definitionof r-functions and theorems involving them. A functional r(x) is called an

r-function if there existsf F withf(x) -< r(x) for all x e E. In (3.11) and (3.12)below, ’ xk stands for the sum x + + x of elements xk e E.THEOREM 3.1. In order that a functional r(x) defined over E may be an r-

function, it is necessary and sucient that

r(tkXk)(3.11) g.l.b. 2_ > 0.n,tk>0; Xk-O k=l tk

THEOREM 3.2.

(3.21) pr(tk Xl)

n,tk>O; ,XkX k=l tk

is a p-function with

r(- x) <- p() (- x) <- p(r) (x) <- r(x);

moreover, if p P and p(x) <- r(x) over E, then

p(’) (x) <- p(- x)

If r(x) is an r-function, then the functional p(r)(x) defined by

(x +E).

R. P. Agnew, On existence of linear functionals defined over linear spaces. This paper isto appear in the Bulletin of the American Mathematical Society.


THEOREM 3.3. In order that p P may have the property p(x) <= r(x) for allx E, it is necessary and su.cient that p(x) <-_ p(r)(X) for all x E.The last theorem follows easily from Theorem 3.2. It is a consequence of

Banach’s Theorem 1.1 that each p-function is an r-function.In 7 we use the following theorem which was not stated in the paper cited

above, but which follows from a slight modification of work in the paper.THEOREM 3.4. Let r(x) be a functional defined over E and let p(r)(X) be defined

by (3.21). If p(r)(X) is finite for at least one x e E, then r(x) is an r-function andp() (x) is a p-function with p(r) (X) =< r(x).

4. Characterization of p-functions having specified properties. In thefollowing and some later theorems, it is possible to replace p0 P by a functionalr(x) not necessarily a p-function and thereby obtain more general theorems.However, existence of f e F with f(x) <= r(x) is, by Theorem 3.3, equivalent toexistence of f e F with f(x) <= p(r)(x); hence we confine ourselves here to p-functions.THEOREM 4.1. Let Po P be fixed. In order that p P may enforce the property

(4.11) f(x) <- po(x) (x E),it is necessary and sucient that

(4.12) p(x) <-_ po(x) (x E).Sufficiency is obvious from our definitions; for, if f(x) <= p(x) and (4.12)

holds, then (4.11) will hold. To establish necessity, let p P and supposex0 e E exists such that p(xo) > po(xo). Let E0 be the linear manifold in Econsisting of elements of the form axo with a R, and let f0 be the linear functionaldefined over E0 by the formulaf0(axo) ap(xo). If a _>- 0, then fo(axo) p(axo);while if a < 0, then f(axo) ap(xo) -p( a xo) <-_ p(- alxo) p(axo).Thus we have fo(x) <-_ p(x) for all x e E0. Therefore Banach’s theorem (loc.cir., pp. 27-28) on extension of linear functionals furnishes a functional f Fsuch that f(x) <-_ p(x) for all x e E and f(x) fo(x) for all x e E0. In particular,f(xo) fo(Xo) p(xo) > po(xo). Thus (4.11) fails and necessity of (4.12) follows.

DEFINITION 4.13. If P0 P enforces a set S of properties, if each p P withp(x) <- po(x) for all x e E enforces S, and if each p e P with p(x) > po(x) for atleast one x e E fails to enforce S, then we shall call p0 the greatest p-functionwhich enforces S.

It is clear that if the greatest p e P which enforces S exists, it is unique. Inthis terminology, Theorem 4.1 states that p0 is the greatest p P which enforces(4.11).THEOREM 4.2. In order that p. P may enforce all properties which p P

enforces, it is necessary and sucient that

(4.21) p(x) <-_ p(x) (x E).

Necessity follows from Theorem 4.1; and su.fficiency is a consequence of thefact that if (4.21) holds, then the class of f e F with f(x) <= p.(x) is a subclass of

6O RALPH PALMER AGNEW

the class of f e F with f(x) <= pl(x). It follows from Theorem 4.2 that twop-functions pl, p. e P enforce the same properties only when they are identical,When properties which some p-functions enforce (or do not enforce) are

known, Theorem 4.2 furnishes a comparison test useful for determination ofproperties which other p-functions enforce (or do not enforce).

Let be (as in 1) a set of pairs {x, y} of elements x, y E having the propertythat if {x, y} e T then/Y, x} e and that {x, x} e T for each x e E.THEOREM 4.3. In order that p P may enforce the property

(4.31) f(y) f(x) (I x, y ),it is necessary and sucient that

(4.32) p[+/- (y x)] 0 ({ x, y ).It is easy to see that Theorem 4.3 follows from the following theorem involving

a single element x0 e E.THEOREM 4.33. In order that p P may enforce the property

(4.34) /(x0) 0,it is necessary and sucient that

(4.35) p(:xo) o.We prove Theorem 4.33. Sufficiency of (4.35) follows from the inequality

p(- x0) =< l(x0)

_p(xo).

To prove necessity, suppose either p(xo) 0 or p(-xo) O. Let 0 :t:x0according as p(:t:xo) O. As in the proof of Theorem 4.1, there exists f Fwith f(x) <- p(x) for all x E and f(0) p(0). Thus f(xo) f(+/-xo):t: f(0) +/- p(0) 0; hence (4.34) fails and necessity of (4.35) follows. Thisproves Theorem 4.33 and therefore Theorem 4.3. On account of our definitionof I,, Theorem 4.3 remains true if we remove the negative sign in (4.32).THEOREM 4.4. Let po P and I, be fixed. In order that p P may enforce

the two properties

(4.41)

(4.42)

each of the four conditions

/(x) <= po(x) (x E),

f(u) =/(x) ({x, y} ),

I ](4.43) p(x) Q.(x) g.l.b, po x + a(y,- x,)n>O; ak R" {xk,Yk} k-..-l

(4.44) p(x) <= Q,(x) g.l.b.n>O; {xk, Yk}

P0 x+-I (yk_xk

n>O; {xk,Yk}e

(4.46) p(x) <- Q)(x) =- g.l.b, po[x -t- (y- x)]{Xl,Yl}


is both necessary and sucient and the condition

(4.47) p(x)<= R(x)-- g.l.b, p0I YIis necessary.

It is essential to observe that Theorem 4.4 does not assert that Q(x),Q)(x) are finite-valued. In many cases (for an example, see 5) one or more ofQa(x), Q(x) is for all x e E; in such cases no p P can exist satisfying(4.43), and it follows that no f e F exists satisfying (4.41) and (4.42). Weobserve also that if I, happens to have the property that {x, y} e I, implies{ax, ay} e for each a R, then Q(x) Q,(x) Qc(x) for all x e E..Since Q(x) <= Q(x) <= Q)(x) and Q(x) <-_ qc(x) _-< Q)(x) for each x E,

we can prove necessity and sufficiency of (4.43), (4.46) by proving necessityof (4.43) and sufficiency of (4.46). To prove necessity of (4.43), let p Penforce (4.41) and (4.42), and let x e E be fixed. Let n > 0; ak R; and let{xk, y} 9. It follows from Theorem 4.3 that

(4.481) p[+/-(y x)] 0 (k 1, ..., n),

hence from Lemma 2.3 that

(4.482)

and therefore from Lemma 2.2 that

(4.483) p(x) plx - .. ak(y x)1.It follows from Theorem 4.1 that p() -< p0() for each e E; and if we letbe the argument of p in the right member of (4.483), we find

(4.484) p(x) <= polx - .. a(y xk)1.From (4.484) we obtain (4.43). To prove sufficiency of (4.46), let p e P satisfy(4.46). Putting y x 0, we see that p(x) <= po(x) and it follows fromTheorem 4.1 that p enforces (4.41). If {x, y} e I,, then we can put y y,x x in (4.46) to obtain

p(x y) <-_ p0[(x- y) - (y x)] p0(0) 0,

and put yl x, xl y to obtain

p(y x) <= p0[(y- x) + (x y)] p0(0) 0.

It follows from Lemma 2.6 that p[(y x)] 0 when {x, y} and hencefrom Theorem 4.3 that p enforces (4.42). This completes the proof of necessityand sufficiency of (4.43), (4.46). To prove necessity of (4.47) we observethat if, in the g.l.b, of (4.44), we require that x x for all/, the g.l.b, will not be


decreased and hence that QB(x) <- R(x).of Theorem 4.4 is complete.THEOREM 4.5. If Po P and

Necessity of (4.47) follows, and proof

Q(x) =- g.l.b, polx - a(yk xk)1n0; ak R; {xk,Yk} 1 kl

is finite for at least one xo e E, then Q.(x) is a p-function (therefore finite-valuedfor all x e E) which enforces the propertiesf(x) <= po(x) andf(y) f(x) for {x, y} e .When x0, x e E; a e R; and {x, y} e , we have

Q(xo) <= po xo - a(y xk po x - a(y x) - (xo x)kl k--l

Hence

Q(xo) po(xo x) <_ po[x k..l ak(yk Xk)1,and we see that finiteness of Q(xo) and po(xo x) implies finiteness of Q,(x).IfxeEandt > 0, then

Q(tx) g.l.b, poltx a(y x)1n0; ak R; {xk,Yk} kl

g.l.b, po x T _, (a/t)(yk- x) tQ,(x).>O; ak/ R; {Xk,Yk} k==l

If x, y e E and > 0 are fixed, we can choose m, al, ..-, am e R andsuch that

end choose , bl, b, R and {x, } e sueh that

0 + 2 b( ) < Q.() + .It then follows from the definition of .(x -t- Y) that

Qa(x W y) <- po (x -t- y) -t- _, a(vl u) + b(y xk)-1 kl

<= Q,(x) W Q,(y) T 2.

Arbitrariness of e > 0 gives

Q(x -t- y) =< Q(x) + Q(y).

It now follows from Lemma 2.1 that Q P and hence from Theorem 4.4that Qa enforces (4.41) and (4.42). This proves Theorem 4.5.


From Theorems 4.4 and 4.5 and Definition 4.13, we obtainTHEOREM 4.6. If Q(x) > for at least one x E, then Q(x) is the greatest

p P which enforces (4.41) and (4.42); if Q(x) - for at least one x E,then no f F exists satisfying (4.41) and (4.42).THEORE 4.7. Let Po P and be fixed. In order that f F may exist with

(4.71) f(x) <= p0(x);

it is necessary and sucient that

(4.72) Q(0) g.l.b.n>0; ak R; {Xk,Yk}

f(y) f(x)

po a(y x O.

To prove necessity, suppose f e F exists satisfying (4.71). Then by Theorem1.3, p e P exists and this enforces (4.71). Then, by Theorem 4.4, p(x) <-so Q(x) must be finite-valued for all x e E. Hence Theorem 4.5 implies Qa e Pand therefore Q(0) 0. This proves necessity of (4.72). Sufficiency of(4.72) follows from Theorem 4.5.THEOREM 4.8. Let Po P and be fixed. If

Q)(x) g.l.b, po[x + (yl- xl)]{Z1,/1}

is finite for at least one xo E, then it is finite for all x E.Proof of Theorem 4.8 is analogous to the first prt of our proof of Theorem

4.5 and is left to the reader. It is easy to show that Q)(tx) tQ)(x) for > 0,x e E in case {x, y} e implies lax, ay} I, for each a e R; and to show thatQD(X - y) <= QD(X) - Q)(y) in case {x, Yl} e and {x2, y} e implies/x W x, y W y} e I,. However, if has these two properties, then Q)(x)Q(x) for all x E.

5. An illustrative example. In this section, let E denote the linear space ofreal bounded functions x =- x(s) defined over < s < , and let

(5.01) po(x) lim x(s) (xE).

Observe that, as the notation implies, p0 e P. Let

(5.02) (s) s-t-(1/12) sin 2rs (- < s < ).

The function q(s) is analytic and has a positive derivative which is bounded andbounded from zero. In fact, 1 /6 =< q(s) -< 1 -t- /6. Finally, let consistof ll pairs {x(s), x(s - )} with x e E, k e R together with all pairs {x(s),x((s)) and {x((s)), x(s)} with x e E. Observe that, for this example, Qa(x)Q,(x) Qc(x) for all x e E. We proceed to show that, for this example,

(5.1) Q(0) g.l.b.n>0; {,y}

P0 (y--x) --.


If M R, Is] denotes the greatest integer =< s, and (s) e E is defined by

(5.21)(s) M (s-[s] < ),

0 (otherwise),

then, since q(s) is increasing, q(s + 1) q(s); (0) 0, and q(1/4) ,(5.22)

(q(s)) M (s-[s] < ),

0 (otherwise).Therefore

(5.23)((s)) (s)

0 (otherwise),

and hence

(5.24) q s+ -- s+ -M (- <s< ),

or

The last equality has the form

(5.25) [y(s) x(s)l -Mkl

where {x, y} e 9. The definition (5.01) of p0 now gives

(- <s< ),

(5.33) f(x(q(s))) f(x(s)) (x E).

It follows that there exists no generalized limit "Lim x(s)", defined for all x E,

having the properties

(5.41) Lim [x() -t- b()] Lim x() -I- b Lim () (,bR;,yN),8--00

(5.42) lim x(s) <- Lim x(s) <= x(s) (xE),

(5.32) f(x(s W X)) f(x(s)) (, R, x E),

(5.26) p0 [x(s) yk(s)] -M.k----1

It follow from (5.26) that the left member of (5.1) is -< -M and, since M R isarbitrary, that (5.1) holds.

Since (5.1) holds, there is no p e P satisfying (4.43) and hence by Theorem4.4 there is no f e F with the properties

(5.31) f(x) <= lim x(s) (x E),

LNEAR FUNCTIONALS AND PRESCRIBED CONDITIONS 65

(5.43) Lim x(s - ) Lim x(s) (k R; x E),

(5.44) Lim x(q(s)) Lim x(s) (x E).

For if such a "Lim" exists we could put f(x) Lim x(s) to obtain f F

satisfying (5.31), (5.32), and (5.33).For the example under consideration we have, when {x, y} I,, either

or

Po(Y x) 1- [=l:x(s W )) :I:: x(s)] _>- 0,

Po(Y- x) =-- l [:i=x(q(s)) =t: x(s)] >= 0.

These inequalities and the equality p0(0) 0 give

QD(0) g.l.b, po(y- x)--O,

and it follows from Theorem 4.8 that Q)(x) is finite for all x e E.From (5.1) and Theorem 4.4 it follows that there is no p e P satisfying any

one of the four conditions (4.43), (4.46). This example therefore showsthat finiteness of Q,(x) does not guarantee existence of f eF with f(x) <- Q)(x),i.e., does not guarantee that Q)(x) is an r-function. In particular, Theorems 4.5,4.6 and 4.7 would not hold if Q(x) were replaced by Q,(x) in their statements.

6. Ftmctionals f e F invariant under G. Let G represent a group of trans-formations g g(x) each of which mapsE univalently into itself and is linear, i.e.,

(6.01) g(ax + by) ag(x) - bg(y) (g G; a, b R; x, y e E).

We shall at times omit parentheses, writing glgx for gl(g(x)), pgx for p(g(x)), etc.Let I, be the set of all pairs {x, g(x) with g e G, x e E. Linearity of g implies

that {ax, ag(x) I, when a e R, g e G, x e E. Thus I, has the property that if{x, y} e , then {ax, ay} e for each a R. Therefore, under the definitions(4.43), (4.44), and (4.45), we have

(6.02) Q(x) Qs(x) Qc(x) (xe E).

These considerations, together with Theorems 4.4, 4.5 and 4.6 give the followingtheorem:THEOREM 6.1. Let po P and G be fixed. In order that p e P may enforce the

properties

(6.11) f(x) <= po(x) (x e E),

(6.12) f(g(x)) f(x) (g G; x E),

66 RALPH PALMER AGNEV

each of the two conditions

[(6.13) p(x) <- Qn(x) g.l.b, p0 x T -1 (gkXk Xkn>0; Ok G; xk E n k=l

(6.14) p(x) <- Q)(x) g.l.b, po[x -b (glxl- xl)]gl G; Xl E

is both necessary and sucient, and the condition

(6.115) p(x) __< R(x) --= g.l.b.p. xn>0; gk G

(xeE)

is necessary. If Q(x) for at least one x E, then no f F exists satisfying(6.11) and (6.12). If Q,(x) > - for at least one x E, then Q,(x) is thegreatest p-function which enforces (6.11) and (6.12).We digress to state and prove a lemma which we shall use in 8.LEMMA 6.2. If F is a subgroup of G; q(x) is a functional defined over E; and

q(x), qa(x) and q4(x) are defined (finite or ) over E by the formulas

(6.21) q(x) g.l.b, q 75 xre>O; 7 P

(.a) () g.l.b.n>O; gk G k=l

then q3(x) q4(x) for all x E.To prove this lemma, we observe first that setting m 1 and 71 I (the

identity of G) in the argument of ql in (6.21) shows that q(x) <= ql(x); it thenfollows on comparing (6.22) and (6.23) that q3(x) <- q4(x).On the other hand it follows from (6.21) and the hypothesis r G that

and hence that

q gk x _>- [_lm] 1] ]g.l.b, q 7; gxm>0; ’i G ;’=I

Taking g.l.b, for n > 0, g e G, and using (6.22), lineurity of the transformationsin G, and finally (6.23), we obtain the estimate,

q(x) > g.l.b g.l.b q1

7igxn>O;Ok G m>0; 7i G

g.l.b, ql1

7lgkX -->__ q4(x).re,n>0; gk,7 q

Combining inequalities gives qa(x) q(x) and Lemma 6.2 is proved.


7. Both p0 e P and f e F invariant under G. Theorem 6.1 does not guaranteethat Q)(x) and R(x), as defined by (6.14) and (6.15), are not -; however,if po(x) is invariant under G, then they must be finite-valued in accordance withTEORE 7.1. If Po P and G are so related that

(7.11)then

(7.12)

po(g(x)) po(x) (g G, x E),

Q(x) g.l.b, po[x 4- (gx x)],gl G; B

(7.13) R(x) ,>o;g’l’b’,. o PE -, gxJare finite-vahedfor all x E and

(7.14) po(- x) Q.(x) po(x)

(7.15) -p0(-x) R(z) po(x)

Setting x 0 in the argument of po in (7.12) shows that Q,(x) po(x).the other hand, if g e G, x E, then (7.11) and Lemma 2.41 give

0 po[gx- x].

0 -< p0[(x -gx- x) x]

<- Po [x+ (gx x)] + P0(-- x),

Hence

On

(7.21) po(g(x)) po(x) (g G; x E).

and po(- x) <- Q)(x) follows. This proves (7.14).Putting n 1 and g I, the identity of G, in the argument of po in (7.13)

shows that R(x) <__ po(x). On the other hand, (7.13) implies

-R(x) 1.u.b. -p0 gxn0; gk G

Using Lemma 2.5, linearity of g e G, and (7.11), we continue to obtain

n)0; gk G kl

so that -po(-X) R(x). This completes the proof of (7.15) and hence ofTheorem 7.1.

In case p0 is invariant under G, we can strengthen Theorem 6.1 by provingthat (6.15) is sufficient to ensure that p enforces (6.11) and (6.12). This factund Theorem 6.1 giveTHEOREM 7.2. Let po P and G be so related that

68 RALPH PALMER AGNEW"

In order that p P may enforce the properties

(7.22) f(x) <= po(x) (x e E),

(7.23)each of the three conditions

f(e(x)) (g G; x E),

is both necessary and sucient.To prove sufficiency of (7.26), let p e P satisfy (7.26). Then (7.26) and

Theorem 7.1 give p(x) <= R(x) <- po(x) so that, by Theorem 4.1, p enforces(7.22). To prove that p enforces (7.23), let g G and x E be fixed. Then(7.26) gives

]p(gx x) <= g.l.b, pol (g gx g, x)

n>0; 0: 0 kl

The g.l.b, on the right will not be decreased if we require that g g-I (go beingthe identity of G) for each k. Hence

(7.27) g.l.b. 1 1,>0

P[gx- x] <-_ glb0. [po(gx) -p0(-x)]

g.l.b.1 [p0(x) T p0(-x)] 0.

n>0 T

Since a similar argument shows that p(x gx) <= O, it follows from Lemma 2.6that p[:t:(gx x)] 0 and hence from Theorem 4.3 that p enforces (7.23).This completes the proof of Theorem 7.2.The condition (7.21) is not sufficient to ensure that Q(x) as defined by (7.24)

is finite-valued. To see this, let E, p0, q, and be specialized as in the exampleof 5. Let G be the group generated by transformations of the form gx(s)x(s - X) with k e R and the transformation gx(s) x(q(s)). Using results of5, it is easy to see that for this example Q,(x) defined by (7.24) is for allx e E. It thus appears that the Q,(x) of Theorem 7.2, which by Theorem 6.1must be a p-function if it is finite-valued, need not be finite-valued. In suchcases no f e F satisfying (7.22) and (7.23) exists; and no p e P satisfying (7.24) or(7.25) or (7.26) exists. Thus we see that Q)(x) and R(x), which must be finite-valued by Theorem 7.1, may fail to be r-functions and hence fail to be p-functionseven when (7.21) holds.


On the other hand it follows from our theorems that if Q,(x) is finite-valued,or if Q)(x) is an r-function, or if R(x) is an r-function, then all must be true andf F exists satisfying (7.22) and (7.23).

It is obvious from the definitions (7.24), (7.25), and (7.26) that Q(x) <-_ Q)(x)and Q,(x) <= R(x) for each x e E. We do not stop here to consider the interest-ing possibility that condition (7.21) may imply either or both of the inequalitiesQ,(x) <= R(x) and R(x)

_Q)(x) for all x E.

The following theorem enables us to add two criteria, namely, p(x) <= Q()(x)and p(x) <- Q(,)(x), to the set (7.24), (7.25), (7.26) in Theorem 7.2.THEOREM 7.3. If po P and G are so related that

(7.31) po(g(x)) po(x)

and Q,(x), Q(I) (x) and Q() (x) are defined for x E by

(7.32) Q,(x)>0; g’l’b’,a:, PIX Tin--- (gx x)l,

(7.33) Q(,)(x)= g.l.b. g.l.b.m>0; Y,xix i=l gi O; i E

(7.34) Q(,’)(x)= g.l.b. g.l.b.m>0; xjx jl nj>O; gik G

then, whether the 5ounds 5e finite or

p0[z + (g-

po gik xi

(g G; x E),

(7.37) Q()(x)- g.l.b. R(x).m>0; Y,xix

(7.36) Q()(x)= g.l.b. Q)(x),re>O; Zxi=x

(7.25) and (7.26),

Since

(7.38) Q,(tx) tQ)(x) R(tx) tR(x), (t > 0; x e E)’

it follows from Theorem 3.3 that (7.25) holds if and only if p(x) <= Q()(x), andthat (7.26) holds if and only if p(x) <= Q(,)(x). Consider now two cases. Incase f e F exists satisfying (7.22) and (7.23), then by Theorems 7.2 and 6.1,Q,(x) is a p-function while Q,(x) and R(x) are r-functions. It follows fromTheorems 7.2 and 3.2 that Q,(x), Q(,)(x), and Q(,)(x) each represent the greatestp-function which enforces (7.22) and (7.23). This gives (7.35) for the first case.In case no f e F exists satisfying (7.22) and (7.23), it follows from Theorems7.2, 6.1, and 3.4 that each member of (7.35) is for all x e E. This completesthe proof of Theorem 7.3.

(7.35) Q,(x) Q(,)(x) Q()(x) (x E).

To prove Theorem 7.3, we observe first that, with Q)(x) and R(x) defined by

7O RALPH PALMER AGNEW

The proof we have just given illustrates the fact that the concept of "greatestp e P which enforces. ." can lead to discovery of equality of functionalswhich might appear to be unequal (or which might possibly be so complicatedthat nothing would be apparent).

8. Both p0 e P andf e F invariant under solvable G. In 7 we saw that R(x),defined by (7.13), may fail to be an r-function and hence may fail to be a p-function even when (7.11) holds. The fundamental result of 8 is that if G issolvable and (7.11) holds, then R(x) must be a p-function. We denote thederived subgroup of G (the subgroup of G generated by the commutatorsglgg-[lg-1) by G’, the derived group of G’ by G", etc. A group is called solvableif some derived group G(r) consists of the identity alone.THEOREM 8.1. If G is solvable and po P has the property

(8.11) pogx pox (g G, x E),

then the functional R(x) defined by

n>o; gk O Ln J

is a p-function.To prove this theorem, let G0 G G G be a sequence of groups

having the properties"1. G0 consists of the identity alone;2. for each 1, 2, r, G_ is the derived group G of G; and3. G is the group G appearing in Theorem 8.1.

With p0 given in Theorem 8.1, we define p(x), p(x), pr(X) by the recursionformulas

(8.13) P(X) g’l’b" P-[l= (=1,2,...,r).

It follows from Theorem 7.1 that p(x) is finite-valued for each 1, 2, ..., rand x e E. Our proof of Theorem 8.1 depends upon two lemmas.LEMMA 8.2. For each 1, 2, r

n>0; gk Ga ’l

The above lemma follows from the definitions (8.13) and iterated use ofLemma 6.2.LEptA 8.3. For each O, 1, 2, r, p(x) is a p-function with the properiy

x)] 0

a if < r, hen p has the property

(8.32) pgx px (g G+; x e E).


Our hypotheses give p0 e P and imply that (8.31) and (8.32) hold when a 0.To complete an induction proof of Lemma 8.3, let 0 < a =< r, p._ P, and

(8.331) Pa-l[’4- (gx

(8.332) p.-lgx p,-lX (g G. x E).

We have seen that p, is finite-valued, i.e., is a functional. We show next thatp, eP.

It follows easily from (8.13) that p.(tx) tp.(x) when > 0 and x e E. Toprove p.(x y) p.x p.y, let x, y e E and e > 0 be fixed. Choose gl, .-.,g, h, h e G. such that

(8.341) p._Fg,x

mkl

Using linearity of the elements of G, the hypothesis p_ e P, and (8.332), weobtain the estimate

p._ hgx p._ h-- gxm ,=1 1 m

< 1 P"-’[h 1 ]=- gx p._ gxn =

But the last member o.f this equality is the left member of (8.341); hence

h(8.351) p._

An analogous estimate together with (8.342) gives

Our hypothesis (8.gal) implies

p.-[(h-- )1 o (h, .; ),

and hence, since ghe elemengs of G map N univalenfly into igself,

p_[(ho h)] 0 (h, .; N).I follows from , h e G, and Lemma 2.8

p_ (h-ih =0.

his equality, Lemma 2.4, and (8.1) imply


Using the definition (8.13) and formulas (8.352) and (8.353), we find

1 ghk(x + y < p,x + p,y + 2.

Arbitrariness of e > 0 gives p,(x - y) <= pax + pay. This proves p e P.If g e Ga, it follows from the definition (8.13) that

and, using (8.332), we can proceed as in (7.27) to obtain pa(gx x) <- O.wise pa(x gx) <= 0 and (8.31) follows.We next prove (8.32). Let g e Ga+l and x e E be fixed.

Lemma 8.2 that

Like-

It follows from

(8.36) pa() g.l.b, p0 g ( e E).n0; gk O

Replacing by x in (8.36), and using (8.11) with g replaced by g-, we find

(8.87) p. g.l.B, p gxn0; ak Oa kl

Since g e G.+ we conclude that g-gg (g- -) , when e G.; hence

(8.88) p.x g.l.b, p

Thus

(8.39) pagx >= p,xIf in (8.39) we replace g by g-1 and x by gx, we obtain

(g Ga+l x e E).

(8.391) pax >--_ pagx (g Ga+; x E).

From (8.39) and (8.391) we obtain (8.32) and proof of Lemma 8.3 is complete.Proof of Theorem 8.1 now follows easily. Definition (8.12) and Lemma 8.2

imply that R(x) pr(x) and Lemma 8.3 therefore gives R(x) pr(x) P.This proves Theorem 8.1.Combining Theorems 7.2 and 8.1 we obtainTHEOREM 8.4. If G is solvable and poe P has the property

(8.41) pogx pox (g G; x E),

then there existsf F with the properties

(8.42) f(x) <- p0(x); (g e G; x e E),and the functional

is the greatest p-function which enforces (8.42).


If G is solvable and (7.21) holds, then, by Theorem 8.1, R(x) is a p-function;it follows from Theorem 7.2 that QB(x), defined by (7.24), must be finite-valuedand hence must be a p-function;and that Q)(x) must be an r-function. Theseobservations and Theorem 8.4 enable us to strengthen the conclusions of Theo-rem 7.3 when G is solvable. Leaving details of proof to the reader, we giveTHEOREM 8.5. If G is solvable and po P has the property

(8.51) pogx pox (g G; x E),

then the three functionals defined for x E by

(8.52) QB(x)=>0; gg’l’b’,o; x, polx _l_n

-1(gx x)l,

(8.53) Q(l)(x) g.l.b. g.l.b, po[xi + g- $i],m>0; xi----x 1=1 Oi O; i

are p-functions (and therefore finite-valued) which are equal for all x e E.Theorem 8.5 is a theorem of analysis which may have some interest apart from

its connection with the theory of linear functionals. We illustrate this remarkby a very special example. Let E be the linear space of real bounded functionsx x(s); let po(x) lim x(s); and let G be the group of transformations of

the form gx(s) x(us W X) where , X e R with > 0. It is easy to verify thatthe hypotheses of Theorem 8.5 are satisfied and it follows that Q,(0) 0; infact Q,(0) 0. It is also easy to verify that the following theorem is merely astatement that Q(0) 0 when E, po and G are as we have specialized them.THEOREM 8.6. U n > 0; x(s), x(s) are real bounded functions;

p > 0; and k h, are real, then

(8.61) lim [x(s T k) z(s)] 0.s

This incidental corollary of Theorem 8.5 may be known; it is given heremerely as a representative of a set of corollaries of Theorem 8.5 which seeminteresting and challenge one to find a simple direct proof.We point out finally in connection with the theorems of 8 involving solvable

groups that the conclusions of Theorems 8.1, 8.4 and 8.5 may hold (at least forsome p0 e P) when G is not solvable. In case pogx pox and p0 happens to belinear, the right members of (8.12), (8.43), (8.52), (8.53) and (8.54) all reduceto the p-function p0, irrespective of the character of G. In this’ trivial case,f e F satisfying (8.42) is obtained merely by setting f p0. It thus appearsthat a necessary and sufficient condition that (8.11) imply that R(x) be a p-function cannot be given in terms of p0 alone or in terms of G alone, but mustinvolve a correlation of p0 and G. In view of this fact, the following theoremmay be of some interest.


THEOREM 8.7. If G (solvable or not) and po P are so related that R(x) P,then for each linear transformation " (not necessarily G) with domain and range inE the functional R(x) defined by

(8.71) R(x) g.l.b, p0 gk ,x (x e E)n>0; Ok G k=l

is a p-function.It follows from (8.43) and (8.71) that R(x) R(,x); hence an application of

Lemma 2.9 gives Theorem 8.7. In case , has an inverse and po.r-lx pox foreach x e E, (8.71) can be written

(8.72) R,(x) g.l.b, p0 / gk ,x (x E).n>0; gk O

Observe that as gk ranges over G, ,-lg, ranges over a group 1 which is simplyisomorphic with G and which is therefore solvable if and only if G is solvable.

9. Applications to limits. Let E denote the set of real bounded functionsx x(s) defined over - < s < oo. As in the example of 1, we write Lim

x(s) in place of a linear functionalf(x). Thus we have a 1-1 correspondence be-tween f e F and linear definitions of Lim, i.e., definitions of Lim for which

(9.01) Lim [ax(s) -b by(s)] a Lim x(s) + b Lim y(s) (a, b R; x, y E).8 8 8

Some additional properties which given definition of Lim x(s) my satisfy(or may fail to satisfy) are

(9.02) g.l.b, x(s) Lira x(s) 1.u.b. x(s) (x E),

(9.03) lim x(s) Lim x(s) lim x(s) (x E),8 8

() d N Lim x() < 1 1 *

h,() ( ),

(9.0) Lim x( + X) Lim () (X R; E),

(9.06) Lim x(us + k) Lira x(s) ( > O;XeR;xeE),

(9.07) Lim x(s) Lim x(s) (, . > O, x E),

(9.08) Lim x(s8- -sin2rs)=Limx(s)- (xeE),

(9.09) Lim x(s - X) Lira x(s) (t, "r > O X e R x e E),

where in (9.04), / and denote respectively lower and upper Lebesgue,integrals.


We leave it to the reader to see what determinations of p0 and G in Theorem8.4 give the following applications.

9.1. The greatest p P enforcing (9.02) is

(9.11) p(x) 1.u.b. x(s).

9.2. The greatest p P enforcing (9.03) is

(9.21) p2(x) l x(s).

9.3. The greatest p e P enforcing (9.03) and (9.05) is

(9.31) p3(z)n>0; Xk R s--, n k=l


(9.41) p(x) g.l.b, lim1

n>0; #k>0;),k R s--, n k=l


(9.151) p(x) g.l.b, li-- 1 1 x(.ke + X)d.,,>o; ,,>o; x,, R o =.


(9.61) p6(x) g.l.b, lim1

n>0;/k>0; ’k>0 s--*

It is easy to give many applications of this sort, and the reader may formulatefurther applications to show even more vividly than the list we have given thatin a sense the greater p-functions enforce fewer properties while the smallerp-functions enforce more properties.

It is of interest to observe that the p-function p3(x) of 9.3, which is the same asp,(x) of (1.23) used by Banach to establish existence of a linear definition ofLim satisfying (9.03) and (9.05), is the greatest peP which enforces theseproperties. There are in fact many different linear definitions of Lim havingproperties (9.03) and (9.05), and (by 9.3 and Definition 4.13) p enforces onlythose properties common to all of them. An example of x0 e E for which it iseasy to show pxo > pxo is Xo(S) sin log+s, where log+s 0 or log s accordingas s _-< 1 or s > 1. The fct that, for each > 0, Xo(S) > 1 over arbitrarilylarge intervals as s - implies that paxo 1. The fact that xo(e’s) -t- xo(s) 0for s > 1 implies that pxo 0. This shows that pa does not enforce (9.06) or,in other words, that not all linear definitions of Lira having properties (9.03) and(9.05) huve property (9.06). On the other hand, property (9.06) obviouslyimplies property (9.05).We illustrate a significance of the notion of "greatest p e P which enforces..."

76 RALPH PALMER AGNENV

by an example. Since p3(sin log+s) 1, there exists a linear functional p7and a corresponding linear definition of Lim with

(9.71) pT(x) Lim x(s) <- p3(x) (x E),

and

(9.72) p(sin log+ s) Lira sin log+ s 1.

Since p e F, we have also p7 e P, and it follows from (9.71) and (9.3) that p7enforces (9.03) and (9.05). Thus pa is the unique greatest p e P which enforces(9.03) and (9.05), while p is merely one of the many p e P which enforce (9.03)and (9.05). We saw in the preceding paragraph that pa is noncommital inregard to (9.06), i.e., enforces neither validity nor failure of (9.06). However,on account of (9.72) and p e F, p actually enforces failure of (9.06).

It was shown in 5 that no linear definition of Lim exists satisfying (9.03),(9.05), and (9.08); hence no p e P exists which enforces these properties.

It appears to be unknown whether a linear definition of Lira exists satisfying(9.03) and (9.09). If such a definition exists then (Theorems 1.3 and 4.4)i5 e P exists with

(9.8) 75(x) =< /(x) - g.l.b, lira 1 x(us. +n>O;lak,/k>O;,k R s--.o kl

so that/(x) is an r-function. On the other hand, if/(x) is an r-function, theni5 e P exists satisfying (9.8). Since obviously/(x) =< p(x), p(x) and p(x) itfollows that/5(x) =< p(x), p(x), and p(x) so that/5 enforces (9.03), (9.06), and(9.07). But (9.06) and (9.07) imply (9.09). Thus we have shown that a lineardefinition of Lim exists satisfying (9.03) and (9.09) if and only if / is an r-function.

10. Application to integration. Let E be the linear space of real functionsx =---- x(s), defined over < s < , for which the upper Lebesgue integral*fix(s) ds of x(s) [over < s < is finite. Let

(10.01) po(x) " x(s) ds (xe E).

Let G denote the group of transformations g such that if g e G, then real and ),

with 0 exist such that

(10.02) gx(s) x(s + )) (- < s < ).It can be verified that g e G is linear, G is solvable, and pgx px for g e G, x e E.

Writing l x(s)ds in place of f(x)for x e E, we obtain a 1-1 correspondence

between f e F and linear definitions of integral, i.e., integrals for which

(10.03) / [ax(s) -+- by(s)] ds a f x(s) ds + b / y(s) ds

when a, b R and x, y e E, and obtain the following application of Theorem 8.4.


10.1. The greatest p P enforcing the properties

(10.11) ,/x(s)ds<- f: x(s)ds <- * x(s)ds (x,E),

x(#s - )) ds(10.12)

is

( O;kR;xE)

p(x)kl

It is easy to make other specializations of p0 and G to obtain other applications,some with greater p-functions which enforce fewer properties and some withsmaller p-functions which enforce more properties.

CORNELL UNIVERSITY.

linear functionals satisfying prescribed conditions

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