3.4.1 Polars and equicontinuous sets
Polars. ConsiderAĂE. Its polar is defined as the set of all functions inE1which have values bounded by1on A:
A˝“ tlPE1|@xPA,|lpxq| ď1u.
Notice thatA˝is absolutely convex andσpEq-closed inE1. Symmetrically one can define the polarB˝ ĂE of a subsetBĂE1:
B˝“ txPE|@lPB,|lpxq| ď1u.
ThusB˝is absolutely convex andσpE1q-closed inE. The following theorem is known as the bipolar theorem and makes the polarization a orthogonality relation:
Theorem 3.4.1. ConsiderAĂE. ThenA˝˝is theσpE1qclosure of the absolutely convex hull ofA:
A˝˝“acxpAqσpE
1q
.
Proof. As a polar is absolutely convex and weakly closed, one hasacxpAqσpE
1q
ĂA˝˝. The converse inclusion is shown by making use of corollary3.3.7.
Equicontinuity.
Notation 3.4.2. Let T be any set of functions f : E //F. Then for A Ă E we denote by TpAq the set tfpxq |f PT, xPAu. ForB ĂF we denote byT´1pBqthe settxPE | Df PT, fpxq PBu.
Definition 3.4.3. A set of functionsK Ă LpE, Fqis equicontinuous if for all0-neighborhoodW Ă F, there exists a0-neighborhoodV ĂEsuch that for alllPK,lpVq ĂW.
Note that equicontinuity does not depend on the topology which may be defined onE1, but only on the one of E.
Example3.4.4. ConsiderUa0-neighborhood inE. ThenU˝is equicontinuous inE1. We recall below the fundamental properties of equicontinuous subsets:
Proposition 3.4.5. [44, 8.5.1 and 8.5.2]
1. An equicontinuous subsetH Ă LpE, Fqis uniformly bounded, that is it sends a bounded set ofE on a bounded set ofF (this amounts to be bounded in the topologyLβpE, Fqdefined in Section3.4.3).
2. The closure H¯ of an equicontinuous subset H Ă LpE, Fq inFE is still contained inLpE, Fq and is equicontinuous.
3. The Alaoglu-Bourbaki Theorem: the closureH¯ of an equicontinuous subsetH ĂE1is weakly* compact.
Proof. ConsiderBa bounded set inEandV a0-neighborhood inF. ThenH´1pVqis a0-neighborhood inE, thus there existsλsuch thatBĂλH´1pVq, and by linearity of functions ofHwe getHpBq ĂλHpH´1pVqq ĂλV. The second point follows immediately from the equicontinuity ofH: one shows the equicontinuity ofH¯ by showing that the reverse image of anyclosed0-neighborhoodV inF is in the0-neighborhoodU inEsuch that HpUq ĂV.
The Alaoglu-Bourbaki theorem follows from the first two points. From2we have that it is enough to show that the closure ofHinKEis compact. But for that it is enough to show that it is a product of compact sets inK, and since compact sets inKare closed bounded subsets the result follows from1.
3.4.2 Boundedness for sets and functions
Working with lcs which may not have a metric, there exists no possibility to define bounded sets as a collection of points which are uniformly at a finite distance from the origin. Instead, one defines a bornology, that is a collection of sets which behaves as a family of bounded sets.
Definition 3.4.6. ConsiderEa vector space. Abornologyon a vector spaceEis a collectionBof setsB called bounded, such thatBis closed by downward inclusion, finite union and coversE.
Unlike continuity, boundedness works with direct images: a function is bounded if the direct image of a bounded set in its domain is bounded in its codomain.
Several bornologies can be defined from the topologyT of a lcs. The most used one is the so-called Von-Neumann bornology:
Definition 3.4.7. A subsetB ĂE is said to beTE-bounded if for every0-neighborhoodU ĂE there exists a scalarλPKsuch that
BĂλU.
This notion of boundedness is compatible with, but not equivalent to, continuity:
Proposition 3.4.8. A linear continuous functionl:E //FisTE-bounded.
Proof. ConsiderBĂEbounded, andUa0-neighborhood inF. Then there existsλPKsuch thatBĂλl´1pUq.
ThuslpBq Ălpλl´1pUqq ĂλU.
Beware that the converse proposition is false. In a non-normed lcs, bounded linear functions may not be continuous.
Example3.4.9. This example is given by Ehrhard in [18]. Consider the vector space`1. It is a sequence space and thus one can define the pfs`1K. Following example3.3.1, there exists a linear isomorphism between`1Kand`8. However as lcs,`1Kis not endowed with the norm8but with the normal topology induced by`KK1 “`8K.
A reasoning similar to the one used in example3.3.1shows that`8K“c0, thus the dual of the pfs`8is the same as the one of`8normed byk¨k8.
Through the Hahn-Banach theorem, one constructs a non-null linear functionf : `8 //Kmapping every sequence inc0 to0, which is continuous fork¨k8. It is thus bounded for the Von-Neumann bornology induced byk¨k8. As`8as a normed space and`8 as the pfs`1Khave the same dual, they have the same bounded sets (see Proposition3.4.10). Thusf is also a bounded linear form on`1K. It is however not continuous for the normal topology induced byc0on`1K: if it were continuous it would be null.
A fundamental property ofTE-bounded sets is that one can test their boundedness scalarly:
Proposition 3.4.10. [44, 8.3.4] A setBisTE-bounded if and only if it is weakly bounded, that is if and only if for everylPE1,lpBqis bounded inK.
Proof. One implication follows directly from Proposition3.4.8. For the converse implication, we will only sketch here the proof of Jarchow done in his chapter 8.3 [44]. It relies on the fact that it is exactly the same property for B to be weakly bounded and forB˝to be absorbent: it means exactly that for everyl P E1, there existsρą 0 such that for everyxP B,|lpxq| ă ρ. ButB˝is also absolutely convex and weakly closed: whenB is weakly bounded, it is thus a barrel (see definition3.4.22). Then the Banach-Mackey theorem [44, 8.3.3] tells us that not onlyB˝absorbs every point ofE1, but it also absorbs globally every Banach diskAofE. This is enough to be able to prove then thatBwill be absorbed by any closed absolutely convex neighborhoodU “U˝˝, asB˝will absorb the Banach diskU˝.
This eases a lot the work on bounded sets of a lcsEas it allows, most of the times, to test a property inKand to infer it inE. For example, a functionf :E //F is bounded is and only if for everylPF1,l˝f is bounded.
Other bornologies can be defined, and each one will be used to define a new topology on spaces of linear functions in Section3.4.3.
Definition 3.4.11. On any lcsEone defines the following bornologies:
• σpEq, the bornology of all finite subsets ofE.
• βpEq, the bornology of allTE bounded sets ofE, also called the Von-Neumann bornology ofE. This is also the bornology of all weakly bounded sets ofE(Proposition3.4.10).
• µpEqis the bornology of all absolutely convex compact sets inEσ, that is of all the weakly compact abso-lutely convex sets. It is called theMackey bornology, and plays an important role in Chapter6.
• γpEqis the set of all absolutely convex compact subsets ofE.
• pcpEqis the set of all absolutely convex precompact subsets ofE.
All these bornologies relates to different notions of duality (Definition3.4.16).
3.4.3 Topologies on spaces of linear functions
LetBa bornology onE. One defines on the spaceLpE, Fqthe topology of uniform convergence on sets ofB, whose basis of0-neighborhood is:
WB,U “ t`PLpE, Fq |lpBq ĂUu forBPBandUa0-neighborhood ofE.
All the bornologies considered in section3.4.2aresaturated, meaning they are stable by bipolar (and thus by absolutely convex closed hull by Theorem3.4.1). IfBis a bornology, then the topologies generated byBor by the setBsat“ tB˝˝|BPBuare the same (see [44, 8.4.1]). Thus one can restrict one’s attention to neighborhoods of the typeWB˝˝,U.
We denote byLBpE, Fqthe vector spaceLpE, Fqendowed with the topologyTB described above. It is a locally convex topological vector space whenFis. The fact that it is Hausdorff follows from the fact that we asked bornologies to coverE.
Proposition 3.4.12. LetEandFbe two lcs, andBa bornology onE. Then the spaceLBpE, Fqis a lcs.
Thus from a bornology on E one can define a topology on LpE, Fq, and in particular on E1 “ LpE,Kq.
Symmetrically, from any bornology onE1, one defines a topology onE:
Example3.4.13. The weak* topology onE1 is the topology of uniform convergence on finite subsets ofE. The weak topologyEwonEis the topology of uniform convergence ofE“LpEw1˚,Kqon finite subsets ofE1.
Note in particular that asE˜1“E1, (see3.1.24) any bornology onE˜ also defines a topology onE1. The strong, simple, compact-open, and Mackey topologies.
Definition 3.4.14. From the bornologies previously constructed in Section3.4one define the following locally convex and hausdorff vector topologies on the vector spaceLpE, Fq:
• the finite bornology onE defines the topologyLσpE, Fqofsimple(sometimes calledpointwise) conver-gence. A filterFĂLσpE, Fqconverges towardslif and only if, for everyxPE,Fpxqconverges towards fpxqinF,
• the Von-Neumann bornology onEdefines thestrong topologyLβpE, Fq, also called the topology of uniform convergence on bounded subsets ofEor thebounded-opentopology,
• Likewise, one defines the lcsLγpE, Fq,LpcpE, FqandLµpE, Fq.
Proposition 3.4.15. On the spaceLpE, Fq,TsďTpcďTµďTβ.
Definition 3.4.16. Each one of these topologies leads in particular to a topology on the dual ofE:
• the topologyEβ1 “LβpE,Kqis called thestrong topology onE, is the most commonly used, and will be studied in section3.5.2,
• the weak topologyE1σpEqwas studied in Section3.3.2,
• the Mackey-topologyEµ1 will be studied in Section3.5,
• the Arens-dual isEγ1, and will be studied in Section3.5.
Definition 3.4.17. Symmetrically:
• the Mackey topologyEµpE1qonE: it is the topology onE of uniform convergence on weakly* compact subsets ofE1.
• the already known weak* topologyEσpE1qonE,
• the strong topologyEβpE1qof uniform convergence on weakly* bounded subsets ofE1.
While a space endowed with its weak topology is called a weak space, a space which is endowed with its Mackey topology is sometime called aMackey spacein the literature.
The equicontinuous bornology onE1defines also a topology onE, but it is exactly the original topologyTE: Proposition 3.4.18. Consider E a lcs. Then TpEq corresponds to the topology of uniform convergence on equicontinuous subsets ofE1.
Proof. We writeTpEEqthe topology onEof equicontinuous convergence on equicontinuous subsets ofE. Then ifUis a0-basis inE, the setsU˝˝form a0-basis ofTpEEqas polars are equicontinuous (example3.4.4). Choosing a0-basis of closed disks one hasU “U˝˝by the bipolar theorem3.4.1and thusTpEq “TpEEq.
Note that finite subsets ofEare in particular weakly-compact, which are in particular weakly bounded, thus bounded (Proposition3.4.10). This leads to the following:
Proposition 3.4.19. ForEandFlcs, we have that the simple topologyTsonLpE, Fqis coarser than the Mackey-topologyTµ, which is coarser than the strong topologyTβ.
The preceding constructions for topologies differ in general, but some coincide on specific subsets ofE:
Proposition 3.4.20. [44, 8.5.1] On every equicontinuous subsetsH ofLpE, Fq, the topologyTS of simple con-vergence coincide with the topologyTpcof uniform convergence on precompact subsets .
Proof. Any finite subset is precompact, therefore the topology of simple convergence is coarser than the precompact-open oneTs ďTpc. Let us prove that the converse is true. Considerl0 P H and let us show that neighborhood ofl0 for the precompact-open topology are in particular neighborhoods for the simple topology. Consider thus a precompact set S Ă E and a closed disk V inUFp0q(thus V “ V˝˝ by the bipolar theorem 3.4.1). As H is equicontinuous, there exists U P UEp0qsuch that for all l P H, lpUq Ă 12V. Since S is precompact, there exists a finite subsetA of H such that S Ă A` 12U. Then if we set M “ 2A, we can compute that pl0`WM,Vq XH Ă pl0`WS,Vq XH.
All these different dual topologies characterize completeness according to a theorem proved by Grothendieck:
Theorem 3.4.21. [51, 21,9.(2)] ConsiderxE2, E1ya dual pair, andBa bornology onE2. Then the completion ofE1endowed with the topology of uniform convergence onBconsists of all the linear functionals onE2whose restriction to setsBPBare weakly continuous.
This theorem relates completion (and thus the possiblity to work with smooth functions), with topologies on dual pairs. It will be used in particular to define chiralities of completeµ-reflexive spaces in Chapter6, section6.4.
3.4.4 Barrels
Barrels are yet another class of subsets of a lcs. The reader which is used to Banach spaces may have never heard of them: they coincide with the balls centred at0sets in normed lcs. They are important in this thesis as they characterize the topology of the double strong dualpEβ1q1β, and thus the reflexivity of a lcsE(see Section3.5.2).
Definition 3.4.22. A subsetU ĂEis abarrelif it is weakly closed, absorbent and absolutely convex. An lcs is said to bebarreledif any barrel ofEis a0-neighborhood.
The following fact was already used in the proof of Proposition3.4.10:
Proposition 3.4.23. A subsetBofE1is weakly* bounded if and only ifB˝is a barrel inE.
Proof. Observe that by definition of boundedness, the polarB˝of a weakly bounded subsetBis absorbent: for allxPE, there existsρą0such that for alllPB,|lpxq| ăρ. ThusxPρB˝. It is absolutely convex and weakly closed because it is a polar, and thus it is a barrel. Conversely ifB˝is a barrel, it absorbs any point ofE, and thus Bis weakly-bounded.
Thus a lcs is barreled if and only if it is endowed with the topology βpE, E1q, that is the topology of uniform convergence on weakly-bounded sets ofE1.
Proposition 3.4.24. [44, 8.5.6] A metrisable complete lcs (that is, a Fréchet lcs)Eis always barreled.
In terms of models of polarized linear logic, barreledness is a positive property. It is preserved in general by inductive limits, and in specific cases by projective limits.
Proposition 3.4.25. [44, 11.3] Barreledness of lcs is preserved by quotient, inductive limits, and cartesian prod-ucts.