Reflexivity

In this section, we will relate the theory of lcs with the fundamental equation of classical Linear Logic, that is the fact that a formula is equivalent to its double linear negation:

A»A^KK.

This corresponds to reflexivity in denotational semantics based on lcs.:

E»E².

Note that the above linear homeomorphisms depends on the topology ofE (which determinesE¹),and on the topology ofE¹(which determinesE²).

Definition 3.5.1. ConsiderBa bornology onE(see Definition3.4.6). A lcs is said to besemi-B-reflexiveif ιE :

#EãÑE^1˚

xÞÑev_x:p`ÞÑ`pxqq (3.5)

defines a linear isomorphismsE „ pE_B¹q¹. It is said to beB-reflexiveif moreover the topologies ofE andE² corresponds, that is ifE» pE_B¹q¹_B.

The restriction linear morphismιEis well defined:evxis continuous onE_B¹.

As the strong topology is the most common topology on the dual, we will by default call semi-reflexivity and reflexivity what is defined above as semi-β-reflexivity andβ-reflexivity. In particular Chapter7deals with strong reflexivity.

3.5.1 Weak, Mackey and Arens reflexivities

In this section we expose some classes of topology on the dual for which semi-reflexivity always holds. We already showed that every lcs is semi-σ-reflexive (Proposition3.3.13):E „ pE_w˚¹ q¹, and that any space endowed with its weak topology isσ-reflexive:Ew» pE_w˚¹ q¹_w˚.

This is also true for the Mackey-topology, whose role is symmetric to that of the weak topology (see the funda-mental Mackey-Arens Theorem3.5.3).

Proposition 3.5.2. Every lcs is semi-µ-reflexive:

E„ pE¹_µq¹, and any space endowed with its Mackey topology isµ-reflexive:

EµpE¹q» pE_µ¹q¹_µ.

Proof. Let us remark that the second linear homeomorphism follows directly form the first one: the Mackey topology onEis the topology of uniform convergence on absolutely convex compact sets inE_σpEq¹ . IfEis semi-µ-reflexive, theσpE¹, EqandσpE¹,pE_µ¹q¹qcompact sets coincide, and thus the Mackey topology induced byE¹ onEcoincides with Mackey topology induced byE¹onE„ pE_µ¹q¹.

Let us prove the first equality. It follows from the fact that pE_σ¹q „ E and from the fact that the Mackey topology is finer than the weak one (every finite subset is indeed weakly compact) that we have a continuous injection:E_µ¹ ãÑE_σ¹. Its transpose results in a linear inclusion:

E„ pE_σ¹q¹ãÑ pE_µ¹q¹. (3.6)

Let us prove that it is surjective. ConsiderφP pE_µ¹q¹. Asφis continuous, there exists a0-neighborhoodV in E_µ¹ such that for every`PV,|φp`q| ă1. By definition of the Mackey topology, this means we have an absolutely weakly* compact subsetK ofE such that we can takeV “ K^˝. This means that if we denote by p¨ ¨ ¨ q^‚ the polar in the dual pairppE_µ¹q¹, E¹,qwe haveφP K^‚‚. Kcan indeed be considered as a subset ofpE_µ¹q¹ through the linear injection. However, thanks to the bipolar theorem3.4.1, we have thatK^‚‚ is the absolutely convex σppE_µ¹q¹, E¹q-closed closure ofKinpE_µ¹q¹. AsKisσpE, E¹q-compact it is in particularσppE_µ¹q¹, E¹q-compact and thusσppE¹_µq¹, E¹q-closed. As is it moreover absolutely convex, we haveK“K^‚‚. AsKis a subset ofE, we obtainφPE.

This proposition implies in particular that for any topology T onE¹ which is finer than the weak topology and coarser than the Mackey-topology, we havepE_T¹ q¹ „ E. This is proved by taking the double duals of the continuous injections:E_σ¹ ãÑE_T¹ ãÑE_µ¹. But we have more:

Theorem 3.5.3. The weak topology onE¹is the coarsest locally convex Hausdorff vector topologyT such that pE_T¹ q¹“E, and the Mackey-topology is the finest.

Proof. LetT be a locally convex Hausdorff vector topology on E¹ such that pE_T¹ q¹ “ E. Then in particular any evalutation functionev_x : E¹ //Kis continuous onE_T¹ , thusT contains all the polars of the finite sets, and is finer than the weak topology. Moreover, we know that the topologyT is also the topology of uniform convergence on equicontinuous sets ofpE_T¹ q¹ “E(Proposition3.4.18). Thus any polarV^˝of a0-neighborhood V inT is an equicontinuous set of E, thus is absolutely convex and weakly*-closed by the Alaoglu-Bourbaki theorem3.4.5. Therefore any closed absolutely convex0-neighborhoodV “V^˝˝ofT is a0-neighborhood for the Mackey-topology. AsT is also generated by its closed0-neighborhood3.1.5, we have thatT is coarser than the Mackey-topology onE¹.

Remark 3.5.4. In particular, any Mackey-dual is endowed with its Mackey-topology. Indeed, aspE_µpEq¹ q¹ „E by proposition3.5.2, we have thatE_µpEq¹ , which is endowed byµpE¹, Eqby definition, is also endowed with its Mackey-topologyµpE¹, E²q. With respect to what is done in the case of weak topologies, the topologyµpE¹, Eq onE¹should be called the Mackey* topology.

We now detail the stability properties of the weak and Mackey topologies:

Proposition 3.5.5. [44, 8.8]

• The weak topology on a projective limit of lcs is the projective limit of the weak topologies.

• The Mackey topology of an inductive limit of lcs is the inductive limit of the Mackey topologies.

• The dual of an injective limit of lcs islinearly isomorphicto the dual of the projective limit.

• [44, 8.8.11] The Mackey dual of a Haussdorf injective limit of lcslinearly homeomorphicto the projective limite of the duals endowed with their respective Mackey topologies.

Theorem3.5.3has an important corollary:

Corollary 3.5.6. IfE_τ¹ is endowed with any topology comprised between the weak topologyσpE¹, Eqand the Mackey-topologyµpE¹, Eq, then the dual ofE¹isE.

Proof. It follows from the hypothesis onEthat we have continuous linear injectionsE_w¹ ãÑE_τ¹ ãÑE_µ¹. Taking the dual of these injections gives linear mapsE //pE¹_τq¹ //E. These maps are surjective by the Hahn Banach theorem, and thus leads to a linear isomorphismpE_τ¹q¹„E.

In particular, as the Arens topology (see Definition3.4.16) satisfies this hypothesis:

Corollary 3.5.7. Any lcsEis semi-γ-reflexive, that is we have the linear isomorphism:pE_γ¹q¹“E.

Thus the Arens dual acts as the Mackey dual or the weak dual in terms of semi-reflexivity. We also have that an Arens dual is alwaysγ-reflexive, meaning that we have a closure operator:

Proposition 3.5.8. For any lcsE, we have

E_γ¹ » ppE_γ¹q¹_γq¹_γ.

Proof. This proof is done for example by Schwartz at the beginning of [67]. We already have the linear isomor-phism by the previous corollary. Let us show that the two spaces in the isomorisomor-phism have the same topology.

Consider any lcsF. Then the topology ofpF_γ¹q¹_γ “ F induces onF the topology of uniform convergence on absolutely convex and compact subsets ofF_γ¹, whileFis originally endowed with the topology of uniform conver-gence on equicontinuous subsets ofF¹. Without loss of generality,Fis also endowed with the topology of uniform convergence on weakly closed equicontinuous subsets ofF¹. But by the Alaoglu-Bourbaki theorem3.4.5, weakly closed equicontinuous sets are in particular weakly compact. As compact sets are weakly-compact, the topology ofpF_γ¹q¹_γ is always finer than the one ofF.

However, whenF “E_γ¹, then equicontinuous subsets ofF¹are by definition generated by the bornologyγpE, E¹q, and thereforeE_γ¹ » ppE_γ¹q¹_γq¹_γ.

3.5.2 Strong reflexivity

In the literature, reflexivity is usually defined with respect to the strong dual: the terms reflexive and semi-reflexive are used for β-reflexive and semi-β-reflexive. Let us describe how β-reflexivity can be understood through a description of the topology of E. The proof makes use of all the notion introduced above, relying on a fine understanding of the role played by the weak and Mackey-topology, and of when a space is endowed with these topologies.

Proposition 3.5.9. [44, 11.4.1] The following propositions are equivalent:

1. Eis semi-β-reflexive,

2. The strong topology and the Mackey topology onE¹coincide:E_β¹ »E_µ¹, 3. Ewis quasi-complete, that is every bounded Cauchy-filter converges weakly.

Proof. p1q ô p2q: As the strong topology is finer than the Mackey topology3.4.15, and by the Mackey-Arens theorem3.5.3, as soon as a spaceE is semi-β-reflexive its dual E¹ is endowed with its Mackey-topology. The converse follows immediately from the preceding Section3.5.1.

p2q ô p3q: IfE_β¹ is endowed with its Mackey topology, it means that every bounded sets inE is weakly-compact, and in particular weakly complete. Conversely, if every bounded setBis weakly complete, from Propo-sition3.1.25it follows that its weak-closure is weakly-compact, and thus that the strong topology is coarser than the Mackey topology. As a weakly-compact set is always weakly-bounded, thus bounded by Proposition3.4.10, the Mackey topology is always coarser than the strong topology and we havep2q ð p3q.

In terms of polarized linear logic, semi-reflexivity is a negative characterization: it is preserved by projective limits, and particular cases by inductive limits:

Proposition 3.5.10. [44, 11.4.5] Semi-reflexivity of lcs is preserved by closed subspaces, projective limits and direct sums.

As semi-β-reflexivity can be modelled through a completeness condition, there exists a closure operator which makes anyE-semi-reflexive: the quasi-completion of Eis semi-β-reflexive and enjoys a functorial property as described in Proposition 3.1.24. Obtaining reflexivity requires much more, and there exists no general closure operator for reflexivity in the context of lcs.

A semi-reflexive lcsE is reflexive if and only ifE carries the same topology aspE¹_βq¹_β, that is the topology of uniform convergence on bounded subsets ofE_β¹. Those are exactly the uniformly bounded subsets ofE¹, that is the sets of functions sending uniformly a bounded set on a bounded set. AsE carries the topology of uniform convergence on equicontinuous subsets ofE¹(Proposition3.4.18), we have:

Proposition 3.5.11. A semi-reflexive space is reflexive if and only if the equicontinuous subsets ofE¹ and the uniformly bounded ones coincide.

In particular, let us recall from Section3.4.4that 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. This topology is exactly the one induced by the strong bidualpE_β¹q¹_βonE. Thus:

Proposition 3.5.12. A semi-reflexive lcsEis reflexive if and only if it is barreled.

Thus reflexivity combines a positive requirement (barreledness, stable by inductive limits) and a negative require-ment (semi-reflexivity, stable by projective limits). This gives some intuition about where the difficulty lies for finding good models ofLLmade of reflexive spaces.

Proposition 3.5.13. [44, 11.4.5] Reflexivity is preserved by cartesian products, direct sums, and strong duality.

ConsiderpE_jqjan inductive system which is reduced (i.e. such that the mapsS_k,jare injective) and such that the E_jare all reflexive. LetE“lim

ÝÑ^jE_jbe the inductive limit of thepE_jq. Then if for every bounded subsetBĂE, there existsjsuch thatBis a bounded subset ofEj, thenEis reflexive.

The following characterization is at the heart of Section7.2.2. From the fact that a metrisable lcs is barreled3.4.24, it follows that:

Proposition 3.5.14. [44, 11.34.3] ConsiderEa metrisable lcs. Then ifEis semi-β-reflexive, it isβ-reflexive and complete.

Outlook4. In a model ofMLL, we will be looking for conditions allowing for reflexivity which are preserved by tensor product. This requires finding a closure operator which makes a space reflexive (to use the Weak, Mackey or Arens dual), or to use a characterization restrictive enough so that it is preserved by some tensor product (as nuclearity, see Chapter7). But even before that, one must choose the topology on the tensor product, and see under which condition this tensor product is associative. This is the content of the next Section3.6.

3.5.3 The duality of linear continuous functions

This short subsection is essential. It provides tools for proving adjunctions of the typeLpE¹, Fq »LpF¹, Eqwhen Eis reflexive for some notion of reflexivity, thus constructing chiralities and models ofMLL. Consider a linear continuous functionf :E //F. Then by precomposition we obtain a linear functionf¹ :F¹ //E¹. For which topologies isf¹continuous ? The next proposition sums up results which are easy consequences of Section3.5.1.

Proposition 3.5.15. [44, 8.6.5]

Considerf : E //F linear. Thenf^˚ :F^˚ //E^˚induces a linear mapf¹ :F¹ //E¹ if and only iff is weakly continuous (i.e continuous with respect to the weak topology onEandF, i.e. continuous with respect to weak* topology onF and any topology onEcompatible with the dualE¹). In that casef¹is weakly* continuous.

Proof. The functionf is weakly continuous if and only iffthe reverse image of the polar of a finite subset ofF¹ contains the polar of a finite subset ofE¹, if and only iff¹maps finite subsets ofF¹to finite subsets ofE¹. Asf mapspE_w¹ q¹topF_w¹q¹we have then by the same reasoning thatf¹is weakly* continuous.

In particular, any continuous linear mapf is weakly continuous, and thus has a transposef¹ : F¹ //E¹. The converse however is not always true: a function can be weakly continuous but not continuous, asFmight contains 0-neighborhoods which are not polars of finite subsets ofF¹. The previous proposition generalizes to:

Proposition 3.5.16. Consider a linear mapf :E //FandBandCbornologies onEandFrespectively. Then f¹:F_C¹ //E_B¹ is continuous if and only if for anyBPBfpBq^˝˝PC.

Let us state very clearly the following easy proposition, which is fundamental throughout this thesis. It could also be deduced from Proposition4.0.10, as reflexive spaces are barreled and thus endowed with their Mackey topology.

Proposition 3.5.17. ConsiderEandFstrongly reflexive spaces. Then we have a linear homeomorphismLβpE_β¹, Fq » LβpF_β¹, Eq.

Proof. Considerf PLpE_β¹, Fq. Asfis continuous, it is weakly continuous, thusf¹:F_β¹ //pE_β¹q¹_βis continuous.

Asfis continuous and therefore bounded, it sends bounded sets to bounded sets, and thusf¹is continuous. AsEis reflexive,f¹is continuous fromF_β¹ //E. The mappingf ÞÑf¹is continuous: indeed, considerB_Fan absolutely convex and weakly closed bounded set inF andBEa bounded set inE. Iff¹pBF˝

q ĎB_E^˝˝, thenfpBE˝

q ĎB_F^˝˝

. The situation being completely symmetrical, we have that forgPLpF_β¹, Eq, we haveg¹PLpE_β¹, Fq, andgÞÑg¹ is continuous.