coursang5.pdf

7/29/2019 Coursang5.PDF

1/110

Part V

Iconoclasm

383


2/110

Chapter 16

Heterodox exponentials

We definitely enter the experimental part of these lectures. From now on, wepresent ideas whose main quality is their radical novelty ; they had thencenot the time to be polished by use : an excess of rigour or details in theirpresentation would be out of place.

16.1 The quarrell of images

One can contend that ludics solves in principle perfect logic. Only in principle :

it provides a setting the analytical theorems where one can perform anot too fabricated synthesis. We are not done, but we can see the end of thetunnel ! The question is to determine what to do with the imperfect part,i.e., with exponentials. The absence of continuity between a perfect world, ofvery restricted expressivity, but harmonious, and an imperfect one where thegrowth of functions cannot be controlled is the sign that something goes wrong.Remember the towers of exponentials whose height is a tower of exponentials. . .do we really believe in that ? This is however the necessary consequence ofthe mental image

of the logical world that we bear in mind ; while the

experience of perfect logic entitles one to question this badly infinite infinity,

this very perennial perenniality. The question is therefore : should one respectmental images, be iconodule, or should one be iconoclast1 ?

16.1.1 Classical absolutism

The strongest iconodule argument is evidence : the world is classical, becauseour fundamental intuitions are classical ; the classical is an absolute that one

1The quarrel of images ravaged Byzantium during the VIIIth and IXth centuries ; theiconoclast enemies of images emperors, like Constantin Copronymos (sic), destroyedall mosaics but in the places they no longer ruled, Ravenna for instance.

384


3/110

16.1. THE QUARRELL OF IMAGES 385

cannot surpass. . . This conformism rests upon a long experience, upon an

undeniable internal coherence ; at the foundational level, it also rests upon amarked taste for essence, for revealed truth.

One must say that even constructivists are of the same opinion : thus,Martin-Lf believes in the set of integers ; should we conclude that his theoryof types is only a dancer, a layer of constructive varnish applied on a classicaland set-theoretic wall ? Everything you just read is nice, but not that serious :when the recess is over, the children must go back to the classical.

What is almost unstoppable. Except that this is circular, that this is thevery blind spot ; you may be right, Mr. Iconodule, but admit that one cannotsee anything : in a case like this, one should only trust indirect evidence.

Fortunately for iconoclasm, there are LLL and ELL, the systems thatwe shall soon introduce. These experimental systems invert to their profit the

revealed

aspect of exponentials, by giving of them lightened versions, with amaintenance of a subtle infinity, inaccessible to set-theorical methods. The soleexistence of these systems is enough to refute the a priori objections, restingupon so-called priority of the classical : while admitting a certain amount ofperenniality, they present it in a less absolute, less desperately frozen, version.

16.1.2 Mathematics

Modifying logic since this is the eventual goal of iconoclasm means givingup mathematics, which can be expressed, as one knows, in set-theory. A lightlogic would thence lose mathematical results : You want to destroy themosaics !

say the iconodules. This is not that obvious :

First, one should not confuse mathematics with mathematics revisited bylogicians, which contain an enormous amount of infinite combinatorics.

Real

mathematics makes a more restricted use of the infinite, and isthence less sensitive to its precise formulation.

If, as in LLL, the function m ; 2m is no longer available, this does notmean that a result mnA[m, n] where the solution n is bounded by 2mis irremediably lost : one could still write it mnA[log m, n]. This ismore complicated, but this removes the objection of principle.

Take a musical analogy : the equal temperament is extremely convenient ;however, the tempered intervals are slightly out of tune. This does not meanthat all music written for the equal temperament but perhaps for certainexaggerations, e.g., dodecaphonism should be dumped. In the same way,classical mathematics is only slightly wrong, as long as one does not enter intologicist exaggerations.


4/110

386 CHAPTER 16. HETERODOX EXPONENTIALS

16.1.3 Sophistics

A sophist is the guy that says to his teacher of sophistics : of two things,either you were a bad teacher and I owe no money to you, or you educated mewell and I can produce a sophism proving that I owe nothing to you

. The same

kind of argument proves the impossibility of motion or the dumbness of generalrelativity, not to speak of quantum mechanics ; if a three-dimensional varietyis embeddable in a four-dimensional Euclidian space, sophistics will concludethat the world is eventually Euclidian : this is the theme ofhyperspace, familiarto readers of science-fiction.

The impressive work done around 1900 enables one to code everything

in set-theory. Hence even the most delirious iconoclasm can be representedin set-theory : which is thence primal, according to a certain sophistics, asin the case of the aforementioned hyperspace. But nobody takes hyperspaceseriously : the Euclidian space is rather seen as a convenient frame for theapproximation of true

geometry. In the same way, one can contend that

set-theory has no real sense, that it is only a convenient reification of a realitydifficult to access.

16.1.4 Iconoclast inconsistencies

The iconoclast viewpoint is delicate, since lacking in coherence ; in particular,

the extant systems, like LLL and ELL are only experimental. But this isa dynamic position, with its future ahead, and a captivating motivation :complexity theory. The gradual setting of an iconoclast logic should lead topose questions in a radically different way. In particular, to find nuances,mistakes, in the prevailing foundational paradigm, which mainly rests on anuncouth approach to natural numbers.

16.2 Exponentials

16.2.1 Kronecker

Any foundational iconoclasm sooner or later stumbles on the absoluteness ofintegers. Thus, the various techniques introduced in these lectures whichmainly belong in finite combinatorics , bring us back to natural numbers.And one does not know how to unscrew

natural numbers. Said Kronecker :

God created the integers, everything else is the deed of man

. How can onecall into question this absoluteness of integers ?

Non standard : Non-standard models of (classical) arithmetic introduce in-tegers after

the true ones

. This is not very convincing : who has

ever seen, who has ever been able to compute, a non-standard integer ?


5/110

16.2. EXPONENTIALS 387

Ultrafinitist : A proposition, not quite serious, by Essenin-Volpin : there

would be integers only up to say 19. This want of earnestness isconfirmed by his claim that these methods can be used to prove. . . theconsistency of set theory ZF !

Dynamic : One no longer poses the question of enlarging or shortening theset N about integers, one ignores it : N is presumably only a reification.When one enunciates a theorem on integers, it is true because it hasbeen proved, which in no way presupposes that {0, 1, 2, 3, . . . } makes anysense. What is important is the process of construction, the dynamics.

16.2.2 A challenge : complexityThe dynamical viewpoint tries to rationalise the intuition that a tower of expo-nentials whose height is in turn a tower of exponential is a pervert effect of theclassical formalism ; that one accepts as a fall-back solution, by no way a real-ity. One cannot not refuse it for a given value of the argument what wouldbring us back to the rut of ultrafinitism , but as a parametric construction.Set-theory does allow the distinction between a parametric construction andthe set of its instanciations corresponding to the values n = 0, 1, 2, . . . of theparameter, i.e., ignores dynamics.

The question became central in the last part of the XXth because of the

emergence of algorithmic complexity. The main achievement of this theory isthe individuation ofcomplexity classes, corresponding to the time or the spaceneeded for the computation. Above all one knows P (problems computable inpolynomial time) and NP (problems verifiable in polynomial time), and thefamous 1000000 $ question :

P?= NP

Each complexity class possesses indirect and interchangeable characterisations,none of them being a mathematical definition in the noble sense of the term what would be for instance a preservation property. Which might be perhaps

related to the rather modest outcome of the area : in more than thirty years,not a sole serious separation result between complexity classe !

I propose to take complexity seriously, not as a problem of informatics,but as a problem of logic. Although iconoclast, the following thesis is veryexciting :

Complexity classes do correspond to various sizes of infinity.

Obviously one is no longer in the Cantorian infinity, which classifies the set-theoretic, static, infinite, in terms of cardinals. One is no more in an inten-sional infinity that would classify the infinite according to stronger or weaker


6/110


systems, such as the reverse mathematics of section 3.C.4 or various bounded

arithmetics , those bleak bureaucracies of complexity. Or, if one prefers, onesurpassed this infantile stage : one will try to play on the only formal trace ofinfinity in logic, exponentials.

16.2.3 Exponentials and integers

And one has one trump card, precisely this abrupt, sharp, modal, side ofexponentials : this is like that

. Which can be read a contrario, since one

can modify them as we please : one will thence explore the possibilities ofmodifications. Everything is permitted, but in order to take only the fair

side of our freedom, one will impose a constraint : the modified exponentialsmust of tame

growth. Polynomial, exponential, this does not matter much,

one will try to tame the growth. This this what will do the lightened logics,LLL and ELL : Light Linear Logic and Elementary Linear Logic firstintroduced in [45].

System F enables one to define integers la Dedekind, remember, sec-tion 6.1.6 : nat := X(X ((X X) X)), that one will write2nat := X((X X) (X X)), what is more legible. This definitiontranslates as nat := X(!(!X X) (!X X)), with a big stock of exclam-ation marks. The same effect would have been obtained with the simplified

version : nat := X(!(X X) (X X)) (16.1)which is the type of the functional sending f XX to f . . .f X X.

Analysing integers, surpassing Kroneckers absolute, this could thence be unscrewing

the !

that struts about definition (16.1). What is not

that simple, since the possible definitions of the exponential deal with finitereuses, finite cliques, i.e., presuppose integers. More generally, one sees thatall analyses based upon categories eventually run into circles. One will searchour happiness with layer 3, with proof-nets. Later, with operator algebras,if possible, see chapter 20.

16.2.4 The lesson of nets

Without much noise, something essential did occur with multiplicative nets.Let us indeed compare the proof of normalisation for natural deduction, sec-tion 4.3.5 with the erroneous proof of normalisation for proof-structures ofsection 11.2.6. In the first case, a parameter, the degree, measures in its waythe logical complexity, thence the algorithmic complexity of normalisation, i.e.,the height of the tower of exponentials of section 4.C.4. In the second case,

2Neglecting the technicalities linked to , see section 7.4.2 !


7/110

16.3. RUSSELLS ANTINOMY 389

on sees that normalisation is performed in linear time, since every step shrinks

the size i.e., the number of links of the structure. What could happen isthat one eventually reaches a vicious circle, but one would then be wise enoughto stop.

Let us go further and think of system F ; there is no longer any degree,since functions grow too fast, but there is still a logical control, operatedby reducibility candidates, see section 6.2.4. On the other hand, if I extendmultiplicative logic to second order, no significant increase in the complexityof normalisation can be observed : the number of steps remains bounded bythe number of links.

To sum up, in the perfect world, logic does not control complexity while

its does in the imperfect world. Could one find a weakened version of theimperfect world lightened exponentials in which one would observe thesame phenomenon, i.e., a priori bounds as to the complexity of normalisa-tion, independent from the logical complexity of formulas, and even from thecorrection of their proofs ?

16.3 Russells antinomy

16.3.1 A paragon of complexity

What is the paragon of algorithmic or logical complexity ? Imagine a veryinexpressive system : for him, there is not much difference between a towerof towers of exponentials and a diverging normalisation, between a systemof high logical complexity and an inconsistent system. In other terms, onecan consider metaphorically non-termination, Russells antinomy, as the worsepossible complexity. Thence the idea of analysing what, in Russells antinomyprovokes non-termination ; one knows that this is due to exponentials. If onecan isolate the principles causing non-termination, and if one can redesignviable exponentials excluding these principles, one will get a logical systemwhose complexity is bounded a priori.

What follows is by many standards very unexpected : in one century, thou-sands of pages have been written on Russells antinomy without any significantprogress. The decomposition perfect/imperfect operated by linear logic enablesone to make a breakthrough on this issue.

16.3.2 Dissection of exponentials

Russells antinomy can be decomposed in two steps :

(i) The construction of a fixed point for negation, say A =!

A.


8/110


(ii) The logical transformation of this fixed point into a contradiction.

The first step is without interest : it means that complexity is not bridled.Everything concentrates on the second step and on the fine grain analysis ofexponentials. Here follows a list of micro-properties

of exponentials :

! (A & B) ! A ! B (16.2)! A ! B ! (A & B) (16.3)A B

! A ! B(16.4)

! A ? A (16.5)

! A ! B ! (A B) (16.6)! A A (16.7)! A ! ! A (16.8)

The first two principles correspond to the isomorphism at the origin of thequalification exponentials

: (16.2) expresses contraction and (16.3) expresses

weakening. (16.4) expresses the functoriality of !

: it is the solution of auniversal problem. (16.5) is a weak form of dereliction. These four principlesconstitute the base of LLL.

(16.6), combined with (16.4), corresponds to multilinear functoriality :from

B conclude !

!B). ELL corresponds to the first five principles.

(16.7) is dereliction.

Burying (16.8) corresponds to the fact that, from! A on deduces ! !A, and not simply ! ! !A. Both principles areseparately faulty : in presence of a fixed point A = ! A, one gets theempty sequent which is not quite contradictory, but not cut-free provableanyway in two ways :

(i) From (16.2) + (16.4) + (16.7)

(ii) From (16.2) + (16.4) + (16.5) + (16.8)

Indeed, A, ?A is an identity axiom ; one deduces ?A, ?A, as we please :(i) By a dereliction (16.7) which directly yields

?A, ?A.

(ii) (16.4) yields !A, ? ?A, then (16.5) yields ?A, ? ?A and (16.8) removesthe extra ?

.

From ?A, ?A, contraction (16.2) yields ?A, i.e., A, which, by (16.4),entails ! A. On concludes by a cut between ?A and ! A.

The principles (16.7) and (16.8) are excluded once for all. The pseudo-dereliction (16.5), which occurs in the reference version [45], has not beenretained here. LLL, with the sole (16.4) would be too weak : what explains the

spare

modality, A, which behaves like a castrated !

, i.e., a supertype

of !

.


9/110

16.4. LLL AND ELL 391

16.3.3 Russell and normalisation

Let us try to understand how cut-elimination works in the two possible trans-lations of Russells antinomy.

The cut between ?A and ! A reduces to (two) cuts between ?A, ?Aand two copies of ! A. Then one proceeds differently :

(i) If ?A, ?A is obtained through a dereliction from A, ?A, one opensone of the boxes ! A which yields A, which yields in turn twocuts between A, ?A, A and ! A. Since A, ?A is an identityaxiom, the cut between A, ?A and A reduces to ?A, whichfinally gives back a cut between

?A and

!

A, the beginning of an

infinite loop of an eternal golden braid, would say the poet !

(ii) In the second case, the system reduces into two cuts between ?A, ? ?A, ! A and ! ! A, next between !A, ? ?A, ? A and ! ! A.Which simplifies into a cut between ? ?A and ! ! A. One loops

too, but by entering one notch inside the boxes

.

One sees that dereliction opens a box (depth 1) to pour it out at depth 0,while burying moves the cut from depth 0 to depth 1. The principles (16.7)and (16.8) which do not respect the depth contrarily to the others arethe cause of non-termination and more generally of unlimited complexity.

16.4 LLL and ELL

16.4.1 The systems

The language contains multiplicatives, first and second order quantifiers, aswell as the exponentials !, ?, , , with A := A, . . . , are useful only inthe case of LLL ; notice the absence of additives. The exponential rules areas follows :

(i) Weakening on all negative formulas (A being declared negative).(ii) Contraction on formulas ?A.

(iii) Two promotion rules :

B1, . . . , Bn, A 1B1, . . . , nBn, B

(16.9)

Either = ! and for i = 1, . . . , n, i =? ; in the case of LLL, n 1.Or =

and for i = 1, . . . , n, i =

or i = ?.


10/110


What is not quite the original system, but rather a simplified variant due

to Asperti [6]. The original system [45] is indeed complicated by the pres-ence of additives ; but, in presence of weakening, they can be defined withoutexponentials by means of :

A B := X ((A X) (B X) X) (16.10)A & B := X ((X A) (X B) X) (16.11)

which are simplified versions of the translations of section 12.B.2. Weakeningis acceptable in a polarised world, see sections 14.C and 15.3.1, which is the

case here.In gneral, there are many variants of ELL and especially LLL, without one

being able to arbitrate in favour of such or such. Thus, the original versionof LLL contained the extra principle (16.5), which was not retained here ; itdeclared the paragraph as self-dual : = ; it had no weakening and wasfussing about a pedantic restriction : exactly one formula in the context of thepromotion of !

. All these variants are based upon the respect of the depth

of boxes, but one hardly sees anything beyond. So, if one is to navigate bysight, the better is to go to the simplest, i.e., Aspertis variant.

16.4.2 Light nets

One uses underlinings like in section 11.C.2 ; but here the only underlinedformulas will be conclusions of boxes, what corresponds to the absence ofdereliction. From a net with conclusions , A3, one can construct a box ofconclusions , !A or , A ; in LLL, the boxes , !A are restricted to the casewhere has at most one element. The link , of conclusion A, has at mostone premise A ; this premise cannot be the conclusion of a link !

. The link?, of conclusion ?A, has n premises A, (n 0). The 0-ary links and ? areparticular cases of a more general

weakening

link, without premise and ofnegative conclusion.

The size is measured depth by depth : s0, s1, s2, . . . . For a certain d,sd = 0, and this degree will not change during normalisation. s0 counts thelinks at depth 0, with a certain pounderation : each link has a weight, thenumber of its non underlined conclusions ; thence boxes, multiplicative andquantifier links count for 1 ; the same for the links , ? and weakening. Butthe axiom counts for 2, while cut is not counted.

3These conclusions are not underlined, what corresponds to the absence of burying.


11/110

16.4. LLL AND ELL 393

16.4.3 Bounds for LLL

One knows that everything normalises, so I will content myself with bounds onthe size of the normalised net. These bounds are easily converted into boundson the computation time.

(i) One begins by normalising at depth 0 all cuts but exponential ones. Oneknows that the size shrinks, so that one can keep the same bounds.

(ii) A cut between two paragraphs, coming from boxes of respective conclu-sions , A et A, , B normalises by burying

the cut at depth 1,between A and A, and by reforming a box of conclusions , , B ;the size shrinks.

Finally, it only remains cuts !/?. These cuts are handled like in the case ofthe paragraph, with an essential difference, due to duplication. If I start with?A0, cut with ! A0, there is a multiplicative factor equal to n 1, where nis the arity of the link of conclusion ?A0. What really causes a problem is thecontext A1 of the box introducing ! A0 : this context is also multiplied byn 1. One sees that the duplication in A0 is transmitted just so to A1 ; onecan proceed in the same way if A1 is premise of a

cut

contraction, what canlead to A2. Indeed, if one wants to count the duplications, one must take thepaths A0, A1, A2, . . . , Ak which are maximal, i.e., which cannot be extended,

neither before 0, nor after k. These paths are finite, i.e., A0 = Ak : it sufficesto choose the adequate switching. How many of them ? This is very simple,as much as possible choices for A0, i.e., less than the size s0 of the net at depth0. Therefore, the size does not increase in depth 0, but it is at most multipliedby a factor s0 in the lower depths. One can redo this at depths 1, then 2, etc.One gets the following bounds :

Profondeur 0 : s0, s1, s2, . . . , sd

Profondeur 1 : s0, s0s1, s0s2, . . . , s0sd

Profondeur 2 : s0, s0s1, s20s1s2, . . . , s

20s1sd

Profondeur d : s0, s0s1, s20s1s2, . . . , s

2d1

0 s2d2

1 . . . s2d2sd1sd

One thence sees that, when normalisation is over, the global size has beenchanged from s = s0 + s1 + s2 + . . . + sd to at most s

2d. For a given d, this apolynomial, thence the polynomial time.

16.4.4 Bounds for ELL

In ELL, exponential boxes are of conclusions , !A, without constraint on .One sees that the cleansing

of depth 0 induces a multiplication by an


12/110


exponential factor of the lower depths, since there are much more sequences

A0, A1, A2, . . . , Ak : the number of such sequences is bounded by 2s0. Onesees that the complexity of ELL, is, for a fixed d, dominated by a tower ofexponentials.

16.5 Expressive power

We know that the complexity of normalisation for a given depth ispolynomial (LLL) or elementary, i.e., bounded by a tower of exponentials(ELL). We shall prove the converse. The essential effort will put on LLL.

16.5.1 Coding of polynomials

In LLL, integers are typed by :

nat := X(!(X X) (X X)) (16.12)One must have an exponential in output

, so as to preserve depth. In ELL,

one can take !

; this does not work with LLL, since one could not typeintegers 2, 3, . . . which require (16.6). This is why one uses the paragraph andthis is by the way the unique reason for its existence.

Given A and f

A

A, one can form !f

!(A

A). If xnat, then

({x}A)!f (A A). In other terms, one can iterate and the result is of type(A A).

One can represent the following functions :

Sum : m, n ; m + n of type nat,nat nat : one basically uses(X X), (X X) (X X).

Product : m, n ; n m of type nat, !nat nat ; one iterates addition, re-written of type !nat!(nat nat), which yields nat, !nat (nat nat),hence nat, !nat, nat nat, corresponding to m, !n, p ; (p + n m).

Square: One can build an object of type nat (nat!nat) corresponding ton ; (n!n). Combining this with the product, one sees that squaringcan receive the type natnat, what corresponds to n ; n2.

General polynomials : No problem : for instance, n4 can be typed natnat.

16.5.2 Coding of polynomial time

One must code integers in base 2 :

nat2 :=

X(!(X

X)

(!(X

X)

(X

X))) (16.13)


13/110

16.5. EXPRESSIVE POWER 395

One can type polynomials in the length

|n

|of a binary integer. For this is

suffices to remark that the identification of the two arguments in a binaryinteger n (by contraction) yields the integer |n| ofnat. The next step is tocode a Turing machine in LLL :

Tape : The tape is can be represented by a generalisation ofnat2, say natN,where N corresponds to the number of symbols that can written on it.

States : A type of the form boolS := X(X . . .XX), with S elements,can be used for the current state of the machine.

Reading head : The type nat can be used for the current position of thereading head.

So that the current position of a Turing machine can be given the typeTur := natNboolSnat, and the machine itself the type TurTur. Giventhe input n nat2, one can type the initial position of the machine pn Tur,as well as the iteration P(|n|) of the machine from the initial position pn.

By cooking all this together, one actually gets that every polynomial timealgorithm can be typed nat2 . . . nat2, with a number of paragraphsdepending on the degree of the polynomial.

16.5.3 Remark on the paragraph

One would search in vain a proof of (A B) A B. I give a proceduralargument ; the same would also work against !(A B) !A!B.

In presence of this principle (supposedly doing what one thinks), every poly-nomial time algorithm with a boolean output (hence of type A B) normallytyped nat . . . (AB), could be retyped nat . . . A . . . B.But normalisation is done in such a way that depth 0 is cleansed

in linear

time. Nothing new will befall later ; now the bit left/right which distinguishesthe two values has been taken back to depth 0. No need to proceed withnormalisation, one already knows the result !

16.5.4 Case of ELL

On defines, without scheming :

nat := X(!(X X) !(X X)) (16.14)and one check that multiplication by 2 can be given the type nat nat.Which enables one to type the iteration of multiplication par 2, a function oftype nat !(nat nat), hence nat, !nat !nat, corresponding ton, !p ; !(p.2n). One thence gets the type nat !nat for the exponentialn ; !2n, hence nat !! . . .!!nat for a tower of exponentials.


14/110

Chapter 17

Quantum coherent spaces

The experimental gait of this last part is easily regressive. Thus, we relin-quished coherent spaces for reason of unfaithfulness, see the discussion insection 12.1 ; this being said, it is a most simple technique which can some-times directly lead to the essential. This is why we shall meet them gain toestablish a link with the quantum world. This link, see [48], is valid for thesole finite dimensional spaces ; to go further, other techniques (Geometry ofInteraction) will be needed, but what we shall see here is worth the dtour.

17.1 Logic and quantic

17.1.1 A missed encounter

No need to go very far to understand why the relation between logic andquantic has been this complete failure : in the same way Frege had the nerveto make fun of the revolutionary ideas of Riemann, logicians were not afraidto declare that nature makes mistakes and therefore to try to reform, to re-format

it. Thence the notorious quantum logic an expression of the style

popular democracy

, where the role of the adjective is to negate the noun1.It is necessary to make it clear from the very start :

What follows has nothing to do with quantum logic

.

17.1.2 Caracteristics of the quantic

What stroke people in the quantic, it its non-determinism, hardly acceptedfor essentially ideological reasons, even by the great Einstein : God doesnot play dice with the world

. However, in a strictly deterministic world,

1Another example : labeled deductive systems ; labeled meaning that the systemis not deductive ; think also of the slogan logic plus control .

396


15/110

17.1. LOGIC AND QUANTIC 397

the theory ofchaos, initiated by Poincar, shows the practical impossibility of

predicting the outcome of the national lottery or the position of the solarsystem in 106 years : this the famous metaphor of the butterfly. But the chaosis not too shocking, since it leaves open the possibility of an abstract, inhumandeterminism : if one actually knew how to photograph the world at instantt, one could deduce its position at instant t + 10. It is however necessary toremark that this if one knew

is just as dubious as the idea that beyond

incompleteness there would bea stable, eternal, notion of truth : chaoslookslike a finite, effective, version of incompleteness, even in some of its readings.

Nothing of the like in the quantum world : one cannot be reassured withthe idea that there would be too many parameters, hardly measurable. One

must admit that measurement influences the result in a very strong sense : notonly it modifies it, but it creates it. Thus, when I measure a spin, I measureit w.r.t. an axis, say, Z, and I find that there is actually a spin 1/2 w.r.t.this axis, while the electron under measurement had none before. This sort ofbehaviour entails non-determinism, what shocked more than one ; it inducesvarious hidden variables

theories, all of them worn out2.

Deeper than non-determinism is the imbrication between the observer andthe system under observation. Contrarily to the usual physical world, thequantic does not accept a dichotomy suject/object.

17.1.3 Quantum logic

Von Neumann himself should be held responsible

for the birth of quantumlogic ; but not guilty, since, in the beginning of the years 1930, it was naturalto make attempts.

It was indeed an approach of level 1 : one modifies the truth values. Oneknows that classical logic admits a semantics in terms of the truth values v, f,and, more generally, in terms of Boolean algebras. Von Neumann did proposeto replace Boolean algebras with the lattice of closed subspaces of a Hilbertspace. . . which yielded strictly nothing. By the way, von Neumann soon took

an otherwise more fruitful direction that of what is known to us as vonNeumann algebras.

Independently from its technical vacuousness, quantum logic was a mistakea priori. Indeed, level 1 rests upon the duality syntax/semantics, i.e., theschizophrenia subjet/object, in opposition to quantum mechanics which restsupon their imbrication. This level of reading supposes a Fregean position :thence the expression the impulsion of M

has a denotation, i.e., a value,

2But in logic : the fashion of quantum computation induced a come back to old moons,style Gleasons theorem , in a new rear-guard attempt at justifying hidden variables andrehabilitate determinism !


16/110

398 CHAPTER 17. QUANTUM COHERENT SPACES

which is a real number ; one can make function the swing which separates to

relate them better the sense and its denotation. But when I say

the spinof e

, this has strictly no denotation, no value

, in no space, over-ornate ornot.

Let us quickly conclude : if an idea is bad, one cannot fix it by a formalisa-tion. This is nevertheless what quantum logicians did by introducing ortho-modular lattices

, thence kicking out the only interesting datum, the Hilbert

space. Since that time, quantum logic vegetates as a theory of lattices, prefer-ably ill-behaved, without any relation to quantum mechanics. Some quantumlogicians dont even know what is a Hilbert space.

The attitude of logic w.r.t. quantic can be summarised by a sophism : one

can describe quantic with mathematics, and mathematics can be embedded inset-theory, hence in logic. Only remains to code this doohickey

. Anything is

good, preferably the most ad hoc possible, to mark a very Fregean reprobationin front of the mistake committed by Nature. One thinks of the anecdotereported by Herodotus (VII,35) : a tempest having destroyed the boat ofships he had set over the Hellespont, Xerxes had the sea whipped. In the sameway, quantum logic is a whipping

of nature, guilty of illogicality

.

17.1.4 The question

Remains anyway the problem of the relation between logic and quantic. Here,more than ever, one sees that the question at stake is precisely that of thechoice of the question. There is a problem, but which one indeed ?

This problem is not that of the logical explanation of quantic, althoughone necessarily expects clarifications. There is a reason, the quantumformalism wave functions, density matrices cannot be replacedwith sets, graphs, in two words with what we meet in the logical world.

A contrario, observe that it is an essentialist prejudice to believe in aworld of ideas a word moreover rather set-theoretic which wouldrule the universe from above. What if the world of ideas were not whatone believes, but were rather quantic ? After all, rather than teachingnature, why not trying to be its pupil ?

Instead of trying to interpret quantic into logic, one will interpret logic intoquantic3.

3This shift of viewpoint took me 30 years of (part time) reflection.


17/110

17.2. PROBABILISTIC COHERENT SPACES 399

17.1.5 Methodology

One must precise what I mean by quantic

. Surely not quantum physicswhich keeps its own life, far from logic ; indeed, I only mean the processusof quantum measurement, the way in which quantum beings interact. Thiscould go as far as trying to integrate the types of quantum socialisation, e.g.,the fermions asocial creatures, like electrons or bosons gregariousparticles in a re-reading of the notion of equality.

One must perhaps consider a third partner, quantum computation, a fash-ionable and interesting idea, albeit still at the stage of science-fiction. Quantumcoherent spaces perhaps bear some relation to quantum computation, and such

a relation would be welcome ; the papers [79] and [95] are anyway encouraging.If not, their theoretical interest lies in a break with set-theory, a break thatcategory-theory sought without finding it. This is why the viewpoint takenhere is the sole opening of the logical space to non set-theoretic techniques.After reading this chapter, it will no longer be tenable to indulge into sophismslike ideas are language, the language being written with symbols a,b,c. . . ;indeed, what if those symbols were not commuting ?

17.1.6 PCS and QCS

The quantic is based upon superposition, just like linear logic which comes fromcoherent spaces, and a sort of superposition principle, remember, section 9.1.1 :ifa =

i ai, then F(a) =

i F(ai) (preservation of disjoint unions). We shall

start from that, with the idea of an analogy cliques/wave functions (rather :density operators), to revitalise level 2.

One proceeds in two steps, first a probabilistic, i.e., commutative

gen-eralisation, then the general case. As to usual coherent spaces, the Ariadnethread is as follows : points do constitute a distinguished basis and cliquescorrespond to subspaces which express diagonally in this base, i.e., with coeffi-cients 0, 1. The probabilistic version allows real coefficients on the diagonal. Asto the quantic version, it steps out from the diagonal

, by allowing, among

others, the introduction of an identity function which cannot be described asa clique, i.e. as a subspace.

17.2 Probabilistic coherent spaces

17.2.1 Desessentialisation

We shall lazily approach the quantic, by first adding a probabilistic, i.e., non-deterministic, dimension. Here, the main reference is the desessentialisation


18/110


of coherent spaces of section 9.1.5. Remember that coherent spaces can be

defined, given a support |X|, by the duality between subsets of |X| :a | b : (a b) 1 (17.1)

and that linear implication corresponds to the adjunction :

((F)a b) = (Sq(F) a b) (17.2)

Non-determinism essentially concerns the connective

: one will ran-domly choose between A and B in A B. One could thence get expressions

a + (1

)

b, with a A, b B. Which suggests replacing cliques with

functions taking their values in [0, 1]. The formulation (17.1) as well as the ad-junction (17.2) will survive this generalisation : typically (17.1) becomes (17.3)infra.

By the way, should I recall it ? We are at level 2, and this has thankyou my God ! nothing to do with fuzzy logic !

17.2.2 The bipolar theorem

Definition 78 (Duality)Let

|X

|be a finite set ; if f, g :

|X

|R, one defines the scalar product

f | g := x|X| f(x) g(x). The positive functions f, g are polar, notationf | g, when :

f | g 1 (17.3)The space R+|X| of all functions from |X| into R+ is thence equipped witha polarity, whose pole is the segment [0, 1], see section 7.1.1. One defines asusual the polar of a set A R+|X|, and :Definition 79 (Probabilistic coherent spaces)A probabilistic coherent space (PCS) is the pair (|X|, X) of a finite support

|X

|and a subset X

R+

|X

|equal to its bipolar.

Theorem 59 (Bipolar)Let X be a PCS ; then

(i) X is non empty ; it indeed contains the null function : 0|X| X.(ii) X is a closed convex set.

(iii) X is downwards stable : iff g X, then f X.Conversely, every subset ofR+(|X|) enjoying (i)-(iii) is a PCS.


19/110

17.2. PROBABILISTIC COHERENT SPACES 401

Proof : Il is immediate that any PCS satisfies (i)-(iii) ; for instance, if f, g

X,

if h X and if0 1, thenf + (1 )g | h = f | h + (1 )g | h 1, hence X is convex.Conversely, if X satisfies (i)-(iii), and if f X, the Hahn-Banach theoremapplied to the real Banach space R(|X|) says that a hyperplane separates X(closed convex) from f. In other terms, one can find a linear form suchthat (X) 1, (f) > 1. This linear form can be identified with an elementh R(|X|), in other terms, (g) = g | h ; one defines h(x) := sup(h(x), 0).Obviously, f | h f | h > 1. If g X, define g(x) := g(x) if h(x) 0,g(x) := 0 otherwise. Since g g, (iii) yields g X. Butg | h = g | h 1, hence h X. Then f X = X. 2

Additives

One adopts here a rather locative viewpoint : one supposes that the supports|X| and |Y| are disjoint. The additive connectives will build PCS with support|X| |Y|. If f R+(|X|), g R+(|Y|), one defines f g R+(|X| |Y|) inthe obvious way, by gluing ; one identifies f with f 0|Y|, g with 0|X| g. Theset :

X& Y := {f g ; f X, g Y} (17.4)is the polar of X Y. On the other hand, X Y is not a PCS ; X Ymust be defined as (X Y), without hope of removing the bipolar. Buttheorem 59 yields :

Proposition 31

X Y = {f (1 )g ; f X, g Y, 0 1} (17.5)Proof : It is enough to remark that the right hand side of (17.5) enjoys condi-tions (i)-(iii) : it is therefore a PCS, moreover the smallest containing X Y,since it is its convex envelope. 2

Multiplicatives

The multiplicative connectives will produce PCS of support |X| |Y|.Definition 80 (Adjunction)If R+(|X| |Y|), iff R+(|X|), one defines ()f R+(|Y|) :

(()f)(y) :=x|X|

(x, y) f(x) = (, y) | f (17.6)

What makes sense due to the finiteness of

|X

|.


20/110


Theorem 60

The function ; () is a bijection between R+(|X| |Y|) and the set oflinear applications from the convex coneR+(|X|) to the convex coneR+(|Y|). can be recovered from the associated linear function = () by means of :

(x, y) = (x)(y) (17.7)

with x(x) := 1, x(y) := 0 for y = x.

Proof : A linear function satisfies (f+g) = (f)+(g) for , 0, andis thence determined by its value on the x, which explains (17.7). Everything

is by the way more or less immediate.2

In the ground case (sets, coherent spaces) this would not work : if and f aresets (characteristic functions), ()f has no reason to be a set. This is why oneintroduced coherence together with its corollary, the unicity of the witness ain equation (8.13).

Definition 81 (Linear implication)IfX, Y are PCS, one defines the PCS X Y of support |X| |Y|, as the setof all such that () sends X into Y.Thus the characteristic function |X| of the diagonal belongs to X X ;indeed (|X|)f = f.

X Y is the polar of {f g ; f X, g Y}, it is why it is a PCS.Which enables one to introduce X

Y := X Y, and dually,X Y = {f g; f X, g Y}.

Proposition 32

is commutative, associative, and distributes over &.

Proof : By introducing the obvious notation

(), one could as well define

X Y as the set of such that () sends Y into X. The result followsby imitation of theorem 51, see chapter 14. 2

17.3 Quantum coherent spaces

It is only a slight exaggeration to say the quantum version corresponds tothe probabilistic one when one has forgotten the base {x; x |X|}. Anyway,the first thing to do is to come back to PCS so as to draw some generalconsiderations.


21/110

17.3. QUANTUM COHERENT SPACES 403

17.3.1 Methodological backlash

PCS have a vague resemblance to the coherent Banach spaces (CBS) of sec-tion 15.A. If one forgets exponentials, one can restrict to finite dimensionalreal spaces. In such a case, a CBS can be handled by means of an Euclidian(i.e., real Hilbertian) space E, by means of the duality :

x | y |x | y| 1 (17.8)

Indeed, if X denotes the unit ball of a normed space on the same E, it isimmediate that the unit ball of its dual can be identified with

X, see

equation (15.8). Hence, any real finite dimensional CBS can be described asa set equal to its bipolar in a appropriate Euclidian space E. This being said,the definition by means of equation (17.8) is slightly more general than CBS ;indeed, ifX = X, x1 := sup {; x X} does not necessarily defines anorm. It is possible that x = 0 or worse, that x = +. What one usuallyhandles by restricting to points of finite norm

, then by quotienting by the

points of null norm

. This eventually amounts at modifying E, which showsthat (17.8) is not quite more general than the definition of CBS. But, if onetakes into account the locative viewpoint, the subtyping X Y means that,on the same vector space E, one can have more coherent objects, i.e., that

the unit ball increases. In other terms, the norm decreases, Y X.It can thence become null, i.e., become a semi-norm ; dually, it can becomeinfinite, and in this case, there is not even a name for what one gets.

PCS are not defined on Euclidian spaces, but on positive cones linked to adistinguished base, what is foreign to the spirit of linear algebra. This beingsaid, positivity can be desessentialised :

Proposition 33f R(|X|) is in R+(|X|) iff for all g R+(|X|) the scalar product f | g ispositive.

Proof : Immediate. 2

In particular, we shall see that the QCS which have however an intrinsicnotion of positivity, positive hermitians call for this variability of positivity.

To sum up, the bilinear form x | y induces three dualities, depending onetakes as pole [1, +1] (which yields the norm, i.e., coherence), [0, +] (whichyields positivity), or the intersection of both, [0, 1], which corresponds to PCS,and what we shall keep just so.


22/110


17.3.2 The bipolar theorem strikes back

Let us go back to theorem 59, in a more general setting. In what follows, E isa finite dimensional Euclidian space. The duality is defined by :

x | y : 0 x | y 1 (17.9)

The problem is the charcaterisation of bipolars.

Theorem 61 (Bipolar)A subset C E is its own bipolar iff :

(i) 0 C.(ii) C is a closed convex set.

(iii) Ifnx C for alln N, then x C.

(iv) Ifx, y C, if, 0 and x + y C, then x C.

Proof : We begin with necessity ; (i) et (ii) are immediate.(iii) : if nx C for n N, and z C, then x | z [0, 1/n] for n N,hence x | z = x | z = 0 [0, 1].(iv) : if z C, then 0 x + y | z 1, 0 x | z, 0 y | z, hence0 x | z 1. (iv) induces a sort of converse to (iii) : if x, x C, thennx + n(x) = 0 C, hence nx C.

Let us now pass to sufficiency, and assume that C satisfies (i)-(iv) ; let C+

be the cone

nN n C (=

R+ C). One can rewrite (iv) :

C = C+ (C C+) (17.10)

If b C, then, by (17.10), one must consider two cases :

b C+ : by Hahn-Banach, there is a d E such that b | d < 0 c | dfor all c C. Condition (iii) implies that I = {c ; n N nc C} is avector space ; | d thence vanishes on I, and one can write C = I C,with C = I C. C is compact : if one embeds E in the projectivespace, C has a compact closure, whose frontier is made of the lines R aincluded in C. But there is no such line (they have been removed andput in I) : the frontier is therefore empty and C compact. It followsthat | d is bounded on C, hence on C, and b | d < 0 c | d .By renormalising d one can suppose that = 1, and then d C andb

C.


23/110


b

C

C+ : the same Hahn-Banach yields d

E such that

p | d 1 < b | d for all p C C+. Suppose that c | d < 0 forsome c C ; then nc C C+ for n N and the values nc | dcannot be bounded by 1. One deduces that 0 c | d 1 < b | d forall c C. As above, d C, and b C.

2

17.3.3 Norm and order

With the notations of theorem 61, in particular C+ := nN n C :

Definition 82 (Domain)The domain FinC of C is the vector space C+ C+ generated byC.Proposition 34FinC = (C ( C)).Proof : If c C, d C ( C), then c | d = 0, which subsists forc FinC, hence FinC (C ( C)). Conversely, if c FinC there is avector d (FinC) such that c | d = 0. But (FinC) = C C ( C),hence c (C ( C)). 2

In other terms, the domain of C is the orthogonal of the

null space of C,that we shall soon define and characterise.

Definition 83The domain FinC is equipped with a semi-norm C and a preorderC :

xC = sup {|x | d| ; d C} (17.11)x C y d C x | d y | d (17.12)

Let =C be the equivalence associated with C.

Proposition 35The kernel0C of the semi-norm C is identical to the equivalence class of0modulo =C.Proof : Obvious. 2

In particular, FinC/0C is a partially ordered Banach space.

Proposition 36(i) C+ is the set of positive elements w.r.t. C.

(ii) 0C = C+

(

C+) = C

(

C).


24/110


(iii) The unit ball w.r.t.

C is(C

C+)

(C+

C).

Proof : (i) and (iii) are respectively the cases b C+

and b C C+

of the proof of theorem 61. (ii) is immediate. 2

A few remarks of a slightly repetitive nature :

(i) The partial order C is continuous w.r.t. the .C : if xn C yn and(xn), (yn) are Cauchy sequence for C with limits x, y, then x C y.

(ii) If 0 C x C y, then xC yC.

(iii) Ifx FinC, then there exist y, zC 0 such that x = yzand y x.

What relation between norm and order for C and norm and order for C ?Nothing new in what follows, it is just a compilation :

Equivalence

x =C y x, y(x = C y x | y = x | y) (17.13)The introduction of the domain FinC, i.e., the fact of considering a partial,

non reflexive, relation, enables this symmetrical formulation.

Positivity

x C+ y(y (C)+ x | y 0) (17.14)The relation C is a preorder on the domain FinC. There is no standard ter-minology for such a relation, seen as a relation on E ; it enjoys weak reflexivity :

x y x x y y (17.15)

The next result generalises the decomposition of a hermitians as a differencesu = u+ u of two positive hermitians :

Theorem 62If x E one can find x+ C+ and x ( C)+ such that x = x+ x andx+ | x = 0 ; this decomposition is unique.

Proof : Let x+ be the projection of x on the convex C, and let x := x x+.One knows that x is the unique y such that y | xy y | z for all z C.This condition is easily transformed into y (C)+ and y | x y = 0. 2


25/110


Semi-norm

xC = inf{ ; y ( C)+ |x | y| y C} (17.16)It is however not the case that |x | y| xC y C for all x FinC,y Fin C.Proposition 37If C D, alors :

FinC FinDC D=C

=D

C DThe last inequality takes its full sense if one modifies the notion of semi-normso as to admit infinite values.

17.3.4 Quantum coherent spaces

Let |X| be a finite dimensional (complex) Hilbert space ; one can apply whatprecedes to E := H(|X|), the space of hermitian operators on |X| ; they arethose operators which are self-adjoint, i.e., such that h = h, i.e. :

h(x) | y = x | h(y) (17.17)

equivalently h(x) | x R. Remember that a hermitian is positive whenh(x) | x R+. Among positive hermitians, all the uu ; indeed everypositive hermitian is of this very form, with u in turn positive, i.e., u =

h.

E is a real vector space, whose dimension can easily be computed : if |X| isof (complex) dimension n, then L(|X|) has the complex dimension n2, hence,as a real vector space, the dimension 2n2. Now every operator can uniquelybe written as 2u = (u + u) + i(iu iu), i.e. as h + ik, with h, k hermitian,which shows that the dimension of the real space H(|X|) is n

2

. This space isequipped with a scalar product (definite positive bilinear form) :

h | k := tr(hk) (17.18)

which makes it Euclidian : tr(hk) = tr(kh) = tr(hk), tr(h2) > 0 for h = 0.Two hermitians h, k are polar when 0 h | k 1.

Definition 84 (Quantum coherent spaces)A quantum coherent space (QCS) of support |X| is a a subset X H(|X|)equal to its bipolar.


26/110


Theorem 61 characterises QCS. One finds below two canonical examples. In

both cases the order relation is the usual ordering of hermitians (h k iffk h is positive) ; on the other hand, according to the case, one gets a normof type sup

or sum

, consistently with the remarks on polarity made insection 15.A.1.

Negative canonical : N is made of the positive hermitians ofnorm 1. N+therefore corresponds to positive hermitians ; on N+, N correspondto the usual norm .

Positive canonical : P is made of the positive hermitians of trace 1. P+therefore corresponds to positive hermitians ; on P+,

P coincides with

the trace norm u1 = tr(uu) ; in this case, h1 = tr(h).Indeed, P = N : one uses |tr(uv)| u v1, and, for h, k 0,tr(hk) = tr(

hk

h) = tr((

h

k)(

h

k)) 0, as well astr(uxx) = u(x) | x.

17.4 Additives

17.4.1 A few reminders on quantum

Some basics of quantum mechanics, limited to finite dimension :

(i) The state of a system is represented by a wave function, i.e., a vector xof norm 1 in a Hilbert space |X|.

(ii) The measurement is represented by a hermitian on |X|. To say thatthe value of x w.r.t. is means that (x) = x. In other terms,there are practically never any value. Worse, if , do not commute,it is likely that they have no common eigenvector, i.e., x cannot have avalue w.r.t. both of and , like in the celebrated uncertainty principle.Thus, the Pauli matrices, see infra, which measure the spin along theaxes X, Y , Z, do not commute : if the spin is +1/2 along axis Z, it iscompletely undetermined along X.

(iii) The measurement process is a Procustuss bed4, which forces the systemto have a value

. In other terms, once the measurement performed, the

wave function x is replaced with an eigenvector x of . This process isnon-deterministic : if|X| is decomposed into a direct sum of eigenspaces

4According to Plutarch, Procustus was a sort of old-style communist, who rubbed outthe differences between men by equating them to the length of his beds ; he was thencestretching the short and shortening the tall : thence the Procustuss bed .


27/110

17.4. ADDITIVES 409

of :

|X

|= |X|, so that x = x, then x

is one of the com-

ponents x, renormalised (multiplied by 1/x), and the probability ofthe transition x ; x/x is x2. This process is the reduction of thewave packet, reduction for short.

(iv) In this approach, wave functions are defined up to a scalar of norm 1.For instance, the spin x of an electron is replaced with its opposite xduring a rotation of angle 2, without affecting the system.

(v) The density matrices (or operators) are due to von Neumann ; they takeinto account the scalar indetermination of the wave function ; above all,they maintain the probabilistic aspect of the measurement process. Adensity operator is a positive hermitian of trace 1. Density operatorsform a compact convex set whose extremal points are of the form xx,where x is a vector of norm 1, i.e., a wave function, transformed into theorthogonal projection on the subspace it generates. During the meas-urement process, xx is replaced with

xx

: this density operatoris a mixture

, a convex combination of the projections xx

/x2,whose coefficients x2 correspond to the probabilities of the possibletransitions.

(vi) This formalism is iterative, i.e., one can measure a density operator, not

necessarily extremal. This means that, if one writes our density operatorh under the matricial form (h) w.r.t. the decomposition |X| =

|X|

(h L(|X|, |X|)), then reduction kills the coefficients h outsidethe diagonal : after measurement, h becomes k = (k), with k =h, k = 0 for = .

(vii) The measurement process is irreversible : ifu ; v by measurement, thentr(v2) tr(u2). If X is of finite dimension n, tr(u2) will therefore vary,trough successive measurements, from the value 1 (extremal point xx),up to 1/n : 1/n I, the tepid

mixture, which conveys no information.

Remark the finesse, the creativity, of this interpretation, compared to whatlogicians may have produced, typically those bleak Kripke models which putside to side arbitrary parallel universes.

17.4.2 Quantum booleans

Booleans form a system with two states, which suggests a two-dimensionalspace, hence 2 2 matrices. The classical viewpoint will soon be breathlessfor reasons of triviality, while the quantum viewpoint which really speaksof the same thing, a two-state system will appear of an incredible depth.


28/110


Commutative booleans

There is little to say if one sticks to the traditional

viewpoint :

(i) The booleans true,false are represented by

1 00 0

and

0 00 1

.

(ii) A diagonal matrix

00

, of trace 1, i.e., such that + = 1, and

positive, i.e., such that , 0, represents a probabilistic boolean. Witha computer-science background, one can even admit + 1, whichcorresponds to a partial probabilistic boolean, true with probability ,

false with probability , undefined with probability 1 ( + ).None of this is that exciting. Now, remember that matrices are operatorswritten w.r.t. a certain base ; and let us imagine that for the reason you prefer say, a travel accident that would have distorted the gyroscopes the basishas been lost

. The booleans are still there, but one can no longer read them !

Then this bleak bureaucracy starts to live ! One is lead to studying hermitiansin dimension 2, which leads to the space R3, not to speak of space-time !

Space-time

Any hermitian writes h = 1/2

t + z x iyx + iy t z

, i.e., t.s0 + x.s1 + y.s2 + z.s3,

with t,x,y,zreal, where the si are the Pauli matrices :

1/2

1 00 1

1/2

0 11 0

1/2

0 ii 0

1/2

1 00 1

(17.19)

The time t is the trace, t = tr(h). Computation of a determinant yields4 det(h) = (t2 (x2 + y2 + z2)), the square of the pseudo-metrics. Also remarkthat tr((t.s0+x.s1+y.s2+z.s3)(t

.s0+x.s1+y

.s2+z.s3)) = tt

+xx+yy +zz.For 1

i= j

3, one gets the anti-commutations si.sj + sj.si = 0.

In order to characterise positive hermitians, remember that any hermitianis diagonalisable : modulo a change of base, i.e., a unitary transformation u,

uhu =

00

, with , R ; h is positive iff , 0. In other terms,

the condition condition det(h) 0 (vectors in position time

) caracteriseshermitians which are either positive or negative. Positivity requires the addi-tional condition : tr(h) 0 : which corresponds to the cone of the future

:

t

x2 + y2 + z2.The most general transformation preserving positivity is of the form

h ; uhu, with det(u) = 1, i.e., u

SL(2) : this is the positive Lorenz


29/110

17.4. ADDITIVES 411

group, which preserves both the pseudo-metrics and the future. While here,

the inverse of u SL(2) is given by :a bc d

1=

d bc a

(17.20)

Hence, inversion extends into an involutive anti-automorphism of the algebraM2(C) of 2 2 matrices. In terms of space-time, this anti-automorphismreplaces the spacial coordinates with their opposites.

The group SO(3) of rotations, which only act on space, corresponds topreservation of time, i.e., of trace. Since tr(uhu) = tr(uuh), one sees that

they are induced by unitary u. In other terms, SO(3) admits a (double)covering by SU(2), the group of unitaries of determinant 1, whose general

form is

a bb a

, with aa + bb = 1. The rotations of axes X, Y , Z and angle

are induced by the eisk , respectively :cos /2 i sin /2i sin /2 cos /2

cos /2 sin /2

sin /2 cos /2

ei/2 00 ei/2

(17.21)

It is obviously a heresy

to divide an angle by 2, since there are two solutions.This is why the covering is double ; this is also why a rotation of angle 2 acts

on the spin seen as a

wave function

as a multiplication by 1. Thiscorresponds to the possibility of replacing u by u in h ; uhu, and thatone cannot continuously choose between both determinations just as onecannot continuously determine a complex square root.

Quantum booleans

Classical

booleans are the orthogonal projections of two 1-dimensionalsubspaces distinguished by the matricial representation. A quantum booleanwill simply be a 1-dimensional subspace. This approach refuses from the startany distinction between true and false : if E is a boolean, its negation is E,

period ! Also remark that, for foreseeable reasons of commutation, it will notbe possible to construct convincing binary connectives.

It remains to determine the 1-dimensional subspaces, i.e., the matricesof orthogonal projections of rank 1. They are the matrices of trace 1 anddeterminant 0, i.e., the points of space-time t.s0 + x.s1 + y.s2 + z.s3, such thatt = 1 and x2 + y2 + z2 = 1, and are therefore in 1-1 correspondence with thesphere S2.

What we just called quantum boolean

is known in physics as the spinof an electron. The measurement of the spin along the axis say Z isgiven by the Pauli matrix s3, whose eigenvalues are

1/2. The value +1/2


30/110


is obtained on the subspace corresponding to1 0

0 0

(spin up alongZ), the

value 1/2 (spin down along Z) corresponding to

0 00 1

.

Probabilistic quantum booleans

The most general case is that of a convex combination of quantum booleans,what corresponds to a positive hermitian of trace 1, a density matrix . Onecan diagonalise a hermitian in appropriate orthonormal base ; in which senseis this unique ?

(i)

1/2 00 1/2

is diagonal in all orthonormal bases ; no form of unicity.

(ii) Outside this case, our boolean writes as b + (1 )c, where b, c areorthogonal booleans and 0 < 1/2, and this, in a unique way.

Reduction occurs when one measures a boolean, what corresponds to the meas-urement of a spin. One must specify a base, which diagonalises the measure-ment operator. One writes our hermitian, quantum boolean, probabilistic or

not, in this base, namely : a bb c. Once the measurement done, it becomes :

a 00 c

, i.e., true with probability a, false with probability c = 1 a, along the

axis true/false corresponding to our base.

Negation

Choosing an orthonormal basis consists in choosing two subspaces of dimen-sion 1, i.e., two quantum booleans et I , whose space-time coordinateswill therefore be (1, x , y , z ) et (1, x, y, z). The vectors A = (x , y, z ) and A correspond to the two possible sense on the same axis (spin up, spindwon). The symmetry w.r.t. the origin corresponds to the anti-automorphisma b

c d

;

d bc a

of the algebra M2(C). This transformation corresponds

to negation. Since symmetry w.r.t. the origin is of determinant 1, it is not inS0(3), and is not induced by an element of SU(2) ; by the way, the elemnetsof SU(2) induce automorphisms, not anti.

Binary boolean connectives

Negation, truly involutive operation, does not involve reduction. Which is nolonger the case for binary connectives :


31/110

17.4. ADDITIVES 413

(i) One cannot combine 1-dimensional projections which do not commute

so as to produce another projection.

(ii) Common sense says that, if one cannot tell the truth from the false, itwill be even more difficult to tell a conjunction from a disjunction.

Hence binary connectives are probabilistic : they yield a probabilistic booleaneven when the arguments are pure

. Moreover, they depend upon a base and

an order of evaluation, for instance :

a bb c

a b

b c

:=

a + ca cb

cb cc

. The

first argument is reduced in the base : true with probability a, in which case the

answer is1 0

0 0

, false with probability c, in which case the answer isa b

b c

.

There is a symmetrical choice, reducing the second argument. And also Jivaro

choice

, which reduces both :

a + ca 0

0 cc

, which is symmetrical, since

a + ca = a + ca = a + a aa.

17.4.3 Additives and quantum

Plus and With

Definition 85 (Additives)IfX, Y are QCS of respective supports|X|, |Y|, one definesX Y andX& Y,QCS of base |X| |Y| :

X Y = {h (1 )k ; h X, k Y, 0 1} (17.22)X& Y = {g ; |X|g|X| X, |Y|g|Y| Y} (17.23)

The subspaces |X|, |Y| have been identified with their orthogonal projections.

Proposition 38

and & are swapped by negation.

Proof : Since h k | h k = h | h + k | k. 2

Neither XY, nor X&Y are norms. This definition mistreats hermitiansthat cannot be written h k. W.r.t. a block decomposition, any hermitian on|X| |Y| writes :

h uu k

, with h, k hermitian. Ifu = 0, this takes an infinite

norm in X Y (if one prefers, is not in FinXY). A contrario, its norm w.r.t.X& Y does not depend on u : the kernel 0X&Y contains all the

0 uu 0.


32/110


Dimension 2

If |X| is of dimension 1, then H(|X|) is of dimension 1 (isomorphic to R) ; thetwo canonical QCS of section 17.3.4 coincide, to yield a QCS 1, correspondingto the segment [0, 1] ofR, ordered and normed naturally .

In dimension 2, H(X) is of dimension 4, with several natural choices :

Spin : the positive canonical. The elements of Spin are the positive her-mitians of trace at most 1, the partial probabilistic quantum booleans

.

Spin : the negative canonical. The elements of Spin are positive her-mitians of norm (in the usual sense) at most 1, the anti-booleans

.

Bool : the of two copies of1. The QCS 1 1 is made of all matricesa 00 c

such that 0 a, c a + c 1, i.e., of partial probabilistic

booleans. It is a subset, a subtype

ofSpin.

Bool : the negation of the previous, i.e., 1 & 1. It is made of the matricesa bb c

such that 0 a, c 1.

Our construction of Bool depends upon a 1-dimensional subspace corres-

ponding to

true , here, the one encoded by Z. Which means that, givena vector A S2, there is a QCS of the booleans of axis A

, noted Bool A.

Proposition 39Spin =

AS2 Bool A.

Proof : Of course, Bool A Spin. Conversely, if h Spin, it can be diagon-alised as

a 00 c

, with 0 a, c a + c 1, w.r.t. a certain orthonormal base

e, f. If A S2 corresponds to e, then h Bool A. 2

Corollary 39.1Spin =

AS2 Bool A.

Reduction : a discussion

I said that we are seeking a logical explanation of the quantic. This being said,what we did enables one to clarify the question of reduction.

In the next section, one deals with multiplicatives, thence with linear im-plication. In particular, one will be able to transform a boolean h Spininto something else, by making use of an element of a QCS Spin

. . ., then


33/110

17.5. MULTIPLICATIVES 415

transform the result by means of another implication. . . Some of these trans-

formations will behave like negation, they will be of

wave style , other like thebinary connectives, will make use of reduction, they will be of particle style

.By the way, one will see that logical operations are rather of the particle

style, with a noticeable exception, the non-etaspansed identity axiom, ratherof style wave

.

In which measure is reduction subjective ? Let us offer us an impossiblehypothesis : let us assume this process of transformation of a boolean com-pleted, i.e., that, in this succession of implications, one was eventually ableto close the system

. Which means that everything ended with a last im-

plication, with values in 1. If I compose all my implications, I see that this

sequence of transformations which eventually

closes the system is nothingbut an anti-boolean k Spin. The result is objective : h | k = tr(hk).But the choice ofk (the transformations, the measurements made on h) is verysubjective. We are not neutral, since on the side of k

. Might as well, since

we stand on the side of k, let us diagonalise it in an appropriate base e, f.

Then h =

a bb c

, k =

00

, and so h | k = a + c. If h =

a 00 c

, then

h | k = h | k, i.e., it is as if we had reduced h.It could be the case that we know that f is a boolean in a certain base

(e.g., if f comes from the measurement of a spin). One selects this base, and

h =

a

00 c

, k =

, and one can then write h | k = a + c. In

this case, one reduced

the observer k into k =

00

in such a way that

h | k = h | k.Finally, reduction is a very subjective matter.

17.5 Multiplicatives

17.5.1 Linear functionalsTheorem 63 (Folklore)Let |X|, |Y| finite dimensional Hilbert spaces. ThenL(L(|X|), L(|Y|)) L(|X| |Y|).

Proof : The complex vector space L(|X|) is generated by rank 1 endomorph-isms : xw(y) := y | wx. If L(L(|X|), L(|Y|)), one defines L(|X| |Y|) by :

(x

y)

|w

z

=

(xw)(y)

|z

(17.24)


34/110


Conversely, given

L(

|X

| |Y

|), if f

L(

|X

|), one defines ()f

L(

|Y

|)

by :(()f)(y) | z = tr( (f yz)) (17.25)

hence () L(L(|X|), L(|Y|)). 2Corollary 39.2If H(|X||Y|), iff H(|X|), then ()f H(Y). The function ; ()is a bijection between H(|X||Y|) and the set of linear functionals from H(|X|)intoH(|Y|).Proof : One easily sees that ()f = (()f), hence a hermitian sends

hermitians to hermitians. Conversely, if is a linear functional from H(|X|)to H(|Y|), then uniquely extends into a C-linear functional from L(|X|)to L(|Y|) : (u) := 1/2((u + u) + i(iu iu)). The C-linear functionalsthence obtained are hermitian, i.e., enjoy (f) = (f), and are therefore inbijection with the hermitians ofH(|X| |Y|). 2

The essential property of () is summarised by the equation :tr((()f) g) = tr( (f g)) (17.26)

Which is reminiscent of stability, introduced in section 8.2.6. It was a matterof linearity in the absence of true linear operations, since one could only use

unions and intersections ; the way to true linearity is marked with the refor-mulations of section 9.1, and we eventually find it again under its literal form,with true sums, true coefficients. Unfortunately, just like Moses passing awayat the gates of Israel, the story of coherent spaces stops here : they will notsurvive, under this ambitious form, to infinite dimension, see section 17.6.1.

17.5.2 A few examples

Example 3If E H(E E) is such that (x y) = y x (the twist ), then

()(xw)(y) | z = (x y) | w z= y x | w z = y | wx | z= (xw)(y) | z

(17.27)

hence ()(xw) = xw. By linearity, ()f = f.

Example 4More generally, let u be a linear map from E to F. Then u u sends E Fto F E, and if EF is the twist from F E to E F, thenU =

(u

u)

H(E

F). It goes without saying that (U)f = uf u.


35/110


Example 5

Let I = E+ F be a decomposition of the identity into orthogonal projections(subspaces). Then R = (E E+ F F) acts as follows :(R)f = Ef E + F f F. R is the typical reduction, which kills

the non-diagonal blocks Ef F and F f E of f.

One can ask what the identity map of E F is doing. One easily sees that(IEF)u = tr(u) IF. Not very inspired. . . But this will be used in our lastexample :

Example 6If E is of dimension 2, then (IEE

E) a b

b c = c bb a , i.e., acts asthe negation. IEE E = 2, where is the orthogonal projection on the

antisymmetric subspace of E E, i.e., the 1-dimensional space made of thex y y x.

17.5.3 Connectives

Definition 86 (Multiplicatives)IfX, Y are QCS, one defines the QCSX Y, of support |X| |Y|, as the setof all sending X into Y :

X Y = { ; f X ()f Y} (17.28)

X Y might as well be defined by :

X Y = { ; g Y g() X} (17.29)

and also as {f g; f X, g Y}. This last expression shows that XYis a QCS. From this, one defines X

Y = X Y andX Y = {f g; f X, g Y}. As usual, is commutative, associativeand distributes over & (up to isomorphism).

Observe that multiplicatives force us to depart from the standard orderingof hermitians. For this, one will suppose that X, Y are positive canonicals, e.g.,X = Y = Spin. Then X Y will consider as positive any hermitian sendingthe (true) positive to the (true) positive. Hence, the twist which behaves asan identity map ; but is a symmetry, a hermitian not at all positive !

Hence X Y is more liberal as to positivity than what one could expect.Dually, X Y is more restrictive, here one is more positive than the King !The positive cone ofXY is the closure of the set of sums i figi, fi, gi 0.Most of (truly) positive hermitians of |X| |Y| are not of this form, typicallythe zz, when z is not a pure tensor.


36/110


17.5.4 and reduction

We already met , see, e.g., section 7.4.2. Can one imagine a bleaker, a morebureaucratic topic ? One would like so much to see this principle wrong ; thenceone observed that ludic faithfulness stumbled grosso modo on this idiocy, seethe discussion in section 14.B.4.

The literature on this topic is depressing. One will make an exception forPlotkin who was the first to refute this doohickey, but in such a fabricatedway, that is the case to say what kills me reinforces me

: at least he was the

first one to do so. But what followed, my God. . . ! A PhD not long enough,at the time when everything is measured in number of pages, of publications,

etc. and thats it :

you redo everything with

. Dear , the providence ofPhD students : 100% transpiration how painful it can sometimes be 0%inspiration.

This dear , precisely, one will brush it the wrong way, not necessarilyonce for all, but in a simple and earnest way. One will, in the case of a

Plus

C = A B, differenciate the native

identity from its etaspansion.Intuitively the difference is very simple : the identity is (x) = x, it recopies anobject without really caring about ; if one thinks of delocation, it is a genuinewavy operation. Etaspansion of the same is based upon the idea that one iseither of type A

or of type B

: the etaspansed identity asks the objectwhether it is of type A or B, and whatever the answer, recopies it identically :it is a cop who asks for your identity papers before letting you pass. Thecommutative, set-theoretic, world cannot separate the real identity from itsetaspansion, the inquisitive

, essentialist, identity ; it leaves no real room

for the potential, reduced to a list of possibilities. In quantum, this is different ;etaspansion is a measuring process followed by a trivial reconstruction : it istherefore a plain reduction.

Without working too hard, let us assume that A, B have dimension 1 sup-ports, and let us consider :

The twist : The generic identity of a space

|C

|of dimension 2. Which writes :

=

1 0 0 00 0 1 00 1 0 00 0 0 1

(17.30)

in any base e e, e f, f e, f f of |C| |C|.

The etaspansed twist : This is the gluing of two identities, that of A andthat of B. W.r.t. a well-defined base, corresponding to the decomposi-


37/110


tion of

|C

|in direct sum, this yields :

=

1 0 0 00 0 0 00 0 0 00 0 0 1

(17.31)

These two things are necessarily distinct : ()

a bb c

=

a bb c

is the true

identity. On the other hand ()

a bb c

=

a 00 c

is only an identity of Procus-

tus

. It is the identity for those matrices which are already of the appropriate

form

a 00 c

. For those which do not fit in this setting (in the essence), it

severs the coefficients outside the diagonal. Indeed, is only the reduction ofthe wave packet, corresponding to the measure of the spin along the axis Z.

In logic, only the identity enjoys

etaspansion. What is not the casehere, since negation can be etaspansed too :

=

0 0 0 00 1 1 00 1 1 00 0 0 0

(17.32)

is such that ()

a bb c

=

c b

b a

; it can be etaspansed into :

=

0 0 0 00 1 0 00 0 1 00 0 0 0

(17.33)

Equivalently () a bb c =

c 00 a : measures the spin along the axis Z,

then swaps it.While here, the matrix represents the double of a projection, the one

projecting on the EPR state

, by the name of a famous paradox imaginedby Einstein to refute quantum non-determinism : it is about the space of thex y y x, a space of dimension 1. Two correlated particles, simultaneouslymeasured at a big distance are supposed to yield opposite spins, in apparentcontradiction with relativity ; but the EPR paradox has been verified. For us,we shall say that it is a Par

, which works as the communicating vessels

of figure 10.2.2 : if one destroys (measures) one side, one finds it again reversed on the other side.


38/110


17.6 Discussion

17.6.1 Infinite dimension

This cannot be convincingly extended to infinite dimension.

(i) The pivot of the construction is the trace, which becomes problematicin infinite dimension. One can define the two-sided ideal of trace-classoperators, but this ideal contains no invertible operator, hence nothinglike the twist.

(ii) Some von Neumann algebras (those of type II1) have a trace. One could

try to define QCS whose support would be of type II1. Restricted toPCS, this would consist in replacing discrete supports with the segment[0, 1], equipped with Lebesgue measure, finite sums becoming integrals.This stumbles on the identity function, which still would be representedby the characteristic function of the diagonal. . . which is of measurezero, hence does not exists

. This impossibility extends a fortiori tothe non-commutative case.

(iii) Technically, the problem is caused by the tensor products of Hibertspaces. One should interpret the identity by something lighter, namelydirect sums : this is what geometry of interaction will do.

The failure of level 2 must be related to the impossibility of a convincingtopological treatment at this level. Although of an infinitely higher quality,level 2 reproduces the basic error of Kripke models, i.e., reduces the potentialto the set of all potentialities. Thus, a proof ofAB which consists in a proofof A and a proof of B, should be represented by a direct sum ; while everyconcrete use combine both proofs in a rather unpredictable way, which requiresa tensor product to represent all possibilities. What does not withstand theinfinite limit, is thence actualisation

, i.e., the reification of possibles, a

common attitude to levels 1 and 2.

17.6.2 Operators vs. sets

The most impressive foundational endeavour of the turn of the XXIth centuryowes nothing to logic : it is the Non Commutative Geometry of Connes, [13].It is an anti-set-theoretic rereading based upon the familiar result :

A commutative operator algebra is a function space.

Typically, a commutative C-algebra can be written C(X), the algebra ofcontinuous functions on the compact X. Connes proposes to consider non


39/110

17.A. INITIATION TOC-ALGEBRAS 421

commutative operator algebras as sorts of algebras of functions over. . . non

existing sets. An impressive blow against set-theoretic essentialism !What does this changes for logic, a priori far astray from considerations

internal to geometry ? I would say that this changes our ideas of finite set, ofpoint, of graph, etc.

The commutative, set-theoretical, world appears as a vector spaceequipped with a distinguished base. All operations are organised in rela-tion to this base, in particular they can be represented by linear functionswhose matrix is diagonal in this base.

The non-commutative world forgets the base ; there is still one, but itis subjective, the one where one diagonalises the hermitian operator oneuses : his

set-theory, so to speak. But, if two hermitians f and g have

non commuting set-theories

, one sees that f+ g has a third set-theorybearing no relation to the previous.

Which enables one to come back for the last time to the unfortunate quantumlogic. Closed subspaces are perfectly enough to speak of whatever we want in particular of the set-theory relative to a given hermitian. But thisdoes not socialise not to speak of the gigantic mistake of orthomod-

ular lattices

, where the Hilbert space has disappeared. Indeed,

0 00 1

et

1/2 1/21/2 1/2

have set-theories

corresponding to the bases { X, Y} and

{2/2( X+ Y), 2/2( X Y)}, but what is the set-theory

of their sum1/2 1/21/2 3/2

? The solution does not belong in lattice theory, it involves the

solution of an algebraic equation, 2 2 + 1/2 = 0. In other terms, the orderstructure of subspaces does not socialise with the basic quantum operation,superposition.

17.A Initiation to C-algebrasWhat follows addresses to the reader that would have forgotten the bases offunctional analysis or to the one that would seek his way through a classicbook on the topic, e.g., [59]. This sort of culture est primal and could byno way be replaced with category-theoretic ruminations loosely inspired fromlinear or operator algebra, e.g., traced monoidal categories.

17.A.1 Hilbert spaces

The most popular Hilbert space is :


40/110


Definition 87 (2)

2 is made of square summable sequences of complex numbers, equipped withthe form :

(an) | (bn) :=n

anbn (17.34)

This is more or less the sole example : indeed, any Hilbert space admits anorthonormal base (ei)iI. Which makes it isomorphic with the space

2(I) ofthe square summable families indexed by I ; and remember Parsevals formula :

x2 =i

|x | ei|2 (17.35)

2 is the most frequent case, corresponding to I denumerable ; the other im-portant case is I finite, what corresponds to this very chapter.

The use of complex numbers comes from the spectral theory, see infra :the spectrum must be non empty (in finite dimension, the characteristic poly-nomial must have a root). From that, the form is not quite bilinear, but onlysesquilinear (sesqui = one and a half) : x | y = x | y. Which is enoughto ensure x | x R (in other terms x | y = y | x).

Finally, the form enjoys x | x > 0 for x = 0. Which yields the celebrated :

Theorem 64 (Cauchy-Schwarz)|x | y|2 x | xy | y, equality occurring only in case of colinearity.Further, 2, equipped with the norm x := x | x1/2 is a Banach space, i.e.,is complete. This being said, x | x 0 is enough in practice : one thenquotients by the kernel of the form ; similarly, if the space is not complete,. . . one completes it ! These two operations are known as the process ofseparation/completion of a pre-Hilbert space.

One can sum Hilbert spaces : the algebraic direct sum H K equippedwith the form :

x

y

|x

y

:=

x

|x

+

y

|y

(17.36)

In particular, x y2 = x2 + y2. The tensor product is obtained byequipping the vector space generated by the formal tensors xy, x H, y Kwith the sesquilinear form defined on pure

tensors by :

x y | x y := x | x y | y (17.37)

The Hilbert space HK is defined as the separation/completion of this pre-Hilbert space. Observe that the tensor product does not factorise bilinearmaps, but only some of them (styled Hilbert-Schmidt) : the form | cannotbe factorised through a linear map from H

H into C.


41/110

17.A. INITIATION TOC-ALGEBRAS 423

17.A.2 The dual space

A Hilbert space is a normed complex vector space, i.e., a Banach space. Itsdual is the set of continuous linear forms, equipped with the norm := sup {|(x)|; x 1}. A Banach which is its own bidual is styled re-flexive, which is an exceptional situation, at least in infinite dimension. Hilbertspaces are the main examples of reflexive Banach spaces. The main tool onBanach spaces is the Hahn-Banach theorem, which takes various forms, e.g. :

Theorem 65 (Separation)If C is a closed convex set in a Banach space and x C, then there is acontinuous linear form and a real number r such that

((C)) r < ((x)).This theorem is already very interesting in finite dimension. In this case, allforms are continuous.

A Banach space E can be equipped with a weakened topology : xi x iff(xi) (x) for all continuous linear forms on E. This topology has anextraordinary property : the unit ball of E is weakly compact. Indeed, theweakened topology is nothing but convergence coefficientwise

. W.r.t.

weakened topologies, there are many compacts sets.

Theorem 66 (Krein-Milman)

Any compact convex set is the closed convex envelope of its extremal border,i.e., of the set of its extremal points.

Remember that an extremal point of the convex set K is a point x Kwhich cannot be written as the barycenter of two other points of the convex.The extremal frontier of a disk is the frontier circle, the extremal frontier ofa convex polygon is its vertices. This theorem applies to density operators,i.e., to positive hermitians of trace 1 (w.r.t the weakened topology when thedimension is infinite). It simply says that any hermitian can be approximatedby barycenters of extremal hermitians by projections of rank 1. What isimplemented in finite dimension by diagonalisation in an orthonormal base

(ei) under the form

1 0 . . . 00 2 . . . 0

. . . . . . . . . . . .0 0 . . . n

; the operator can thence be writtenas the barycenter of the projections associated with the ei with the respectiveweights i.

Some Euclidian geometry : let us consider the triangle of vertices 0, x , y andthe median starting with 0, in other terms the vector (x + y)/2. An immediatecomputation yields :

x

2 +

y

2 = 2(

(x

y)/2

2 +

(x + y)/2

2) (17.38)


42/110


This equality enables one to majorise, in certain cases, the norm of the third

edge, x y. Thus, ifE H is a non-empty closed convex set, (17.38) enablesone to project on the convex E : indeed, if(xn E) is such that xn convergesto inf{x; x E}, then it is a Cauchy sequence.

Proposition 40The minimum inf{x ; x E} is reached in a unique point of E. This pointis also the unique e E such that e | e f is negative for all f E.

What applies, mutatis mutandis to the projection of an arbitrary point ona closed subspace E ; hence, let the map obtained in this way. One sees

that is linear and that 2 = ; the range of is E, its kernel is E, andthe projection associated with E is I . These spaces are supplementary,i.e., each vector ofH uniquely writes as x = e + e, e E, e E, i.e., H isisomorphic to E E.

If e H, x x | e is a continuous linear form : by Cauchy-Schwarz|x | e| ex, equality being reached with x = e, hence e = e.Conversely, any linear form continuous on H is of the form e for a well-chosen e necessarily unique : it suffices to consider {x; (x) = 1} and applyprojection techniques. Hence the dual H ofH is canonically isomorphic to Hby means of the map b b ; but, pay attention to the scala

coursang5.pdf

Documents