Multiplicative connectives

6.4.1.1 An internal hom-set on convenient sets

This Section reviews results which can be found in [44, 13.1] or [26].

Proposition 6.4.1. IfE is convenient (i.e. bornological and Mackey-complete), thenE is endowed with the topologyβpE, E¹qof uniform convergence on weakly bounded subsets ofE¹(i.e on subsetsBsuch that for every xPE Bpxqis bounded inK). That is, a convenient lcs is barrelled , see Section3.4.4.

Proof. IfEis bornological it is linearly homeomorphic to the inductive limit of theEB, whereBis an absolutely convex and weakly closed bounded subset ofE (see Proposition6.2.6). WhenE is Mackey-complete theseEB

are Banach spaces by definition. However a Banach spaceEBis always barrelled: it is as all lcs endowed with the topology of uniform convergence on equicontinuous subsets ofE¹_Band these are exactly the simply bounded sets, due to Banach Steinhaus theorem. Moreover,EB has its Mackey-topology, and the Mackey-topologyµpE, E¹q is preserved by inductive limits [44, 8.9.11]. ThusE as a bornological space is endowed with the topology µpE, E¹q “ indBµpEB, E_B¹ q “ indBβpEB, E¹_Bq. As a weakly bounded set in E¹ is in particular a product of weakly bounded subsets ofE_B¹ , and weak topologies are preserved by projective limits [44, 8.8.6], we have our result.

Thus, as a bornological space is always endowed in particular with its Mackey-topologyµpE, E¹q(see Propo-sition6.2.6), we have that through the bipolar theorem:

Proposition 6.4.2. WhenEis a convenient space, the bornologyµpE¹qof absolutely convex and weakly compact subsets ofE¹ and the one of absolutely convex and weakly closed bounded sets (i.e. of the bipolars of weakly bounded sets inE¹) coincide.

From the fact that the space of linear bounded maps to a complete lcs from another lcs is Mackey-complete6.2.19, we have immediately:

Proposition 6.4.3. IfEis bornological andF is Mackey-complete, thenLβpE, Fqis Mackey-complete for any lcsE.

From Proposition6.4.2, we have thus:

Proposition 6.4.4. IfEis convenient andF is Mackey-complete, thenLµpE, Fqis Mackey-complete for any lcs E.

In particular, when we bornologise the hom-set we obtain a convenient space:

Corollary 6.4.5. IfEandF are convenient vector spaces, the lcsLbornpE, Fqis also convenient.

This results hints for a good chirality between convenient spaces (for the positives) and Mackey-complete spaces (for the negatives). In fact, in order to ensure a good behaviour of the Mackey duals (for the negation), we will need to consider complete and Mackey lcs as interpretation for the negatives, see Theorem6.4.16.

6.4.1.2 A duality theory for convenient spaces

We showed in Proposition6.4.4that the space of linear maps between a convenient space and a Mackey-complete space is Mackey-complete. When we consider linear scalar forms, we have a stronger result which appears for example in [44, 13.2.6]:

Proposition 6.4.6. IfEis bornological, thenE¹_βis complete. Thus ifEis bornological and Mackey-complete,E_µ¹ is complete.

Proof. The second assertion follows from the first by using Proposition6.4.2. The first assertion is straightforward.

Consider` PEĂ<sub>β</sub><sup>1</sup>. It is linear as the pointwise limit of linear map. It is bounded: considerBbounded inE, then there is`nPE¹such thatp`´`nq PB^˝, thus`pBq Ă`_pBq `B0,1whereB0,1is the unit ball inR. Thus`pBqis bounded inRfor anyBbounded, thus`is bounded and thus continuous asEis bornological. thusE˜¹β“E_β¹.

Recall that theεproduct (see Section3.6.3), defined asEεF :“L_εpE_γ¹, Fqis associative and commutative on complete spaces.

Proposition 6.4.7. WhenEandF are convenient spaces, then we have a linear homeomorphismpEbβFq¹_µ » E_µ¹εF_µ¹.

Proof. We have the following computations:

pEbβFq¹_β»BpEˆF,Rqas E and F are bornological

»LpE, F^ˆq

»LβpE, F_β¹qasEandFare bornological

AsE andF are convenient, we haveLβpE, F_β¹q » LµpE, F_µ¹q(Proposition6.4.2). Moroever any weakly compact set inE_µ¹ is compact asE_µ¹ is endowed with its Mackey topology (see [68]), and thuspE_µ¹q¹_γ » pE_µ¹q¹_µ. Finally, the equicontinuous subset ofEseen as the dual ofE_µ¹ are exactly the subsets ofµpE, E¹q: equicontinuous subsets ofE are polarsU^˝ of open sets inE_µ¹, and openU sets inE_µ¹ are by definition polarsW^˝ of weakly compact and absolutely convex subsets ofE. By the bipolar theorem3.4.1, equicontinuous subset ofEare exactly the subsets ofµpE, E¹q, and thusLµpE, F_µ¹q »LεpE, F_µ¹q » pE_µ¹qεpF_µ¹ by the preceding reasoning.

We will also make use of Lemma 3.7 of [17], which relates Mackey-completions and Mackey-duals. We recall it below:

Proposition 6.4.8. ConsiderEa space endowed with its Mackey-topology. ThenEˆ^M is still endowed with the Mackey-topologyµpEˆ^M, E¹q.

Proof. Remember that our Mackey-completion preserves the dual, thusµpEˆ^M, E¹qis indeed the Mackey topology onEˆ^M. MoroeverEˆ^M is constructed as the intersectionEp^M of all Mackey-complete spaces containingE and contained in the completionE˜ ofE. Therefore an absolutely convex weakly compact set inF¹coincide for the weak topologies induced byF andFr and therefore alsoFˆ^M, which is in between them. Thus the continuous inclusionsppF_µ¹q¹_µq //pFˆ^Mq¹_µq¹_µ //ppFqr ¹_µq¹_µhave always the induced topology. In the transfinite description of the Mackey completion, the Cauchy sequences and the closures are the same inppFrq¹_µq¹_µandFr(since they have same dual hence same bounded sets), therefore one finds the stated topological isomorphism.

We adapt now a proof which can be found in [51, 28.5.4] or [44, 13.2.4]. The same generalisation can be found in the unpublished Master’s report [27] by Gach, directed by Kriegl.

Proposition 6.4.9. LetEbe a Mackey lcs such thatE¹_µis complete. ThenEis bornological.

This uses Grothendieck’ characterization of completeness (Theorem3.4.21).

Proof. LetEbe a Mackey space whose strong dual is complete. By proposition6.2.6we just have to show that E¹ “ E^ˆ,i.e. that any bounded linear function onE is continuous. Considerf P E^ˆ, let us show thatf is continuous. We make use of Grothendieck’s Theorem3.4.21, which says thatE¹_µ»E˜_µ¹ „ tf :E //K| @KĂ E, Kabs. convex weakly compactf_|K : K_σpE,E1q //Ku is continuous. Thus considerK a weakly compact

absolutely convex subset ofE. We consider the normed spaceEK, normed withpK(see Definition3.1.12). AsK is in particular bounded, we have thatf is bounded and thus continuous on the normed spaceEK endowed with pK.

However onK ĂE the weak topology andpK coincide. Indeed the weak topology is coarser than the one induced by pK onEK (that is the intersection betweenEK and a weak-open set is an open set forpK),K is compact inEKendowed withpK, and thus onKĂEthe weak topology andpKcoincide [51, 28.5.2].

Thus onK pK and the weak topologyσpE, E¹qcorrespond, and a bounded linear functionf is in particular linear continuous onK_σpE,E1q. ThusE^ˆ„E¹.

Corollary 6.4.10. LetFbe a Mackey lcs which is complete. ThenF_µ¹ is bornological.

Proof. Let us denoteE:“F_µ¹ the Mackey-dual ofE. ThenE_µ¹ »F asF is Mackey. Moreover,E_β¹ is endowed with a topology which is finer (any weakly compact absolutely convex set is in particular weakly bounded and thus bounded by Proposition3.4.10). Thus asE_µ¹ is complete, so isE¹_β. Thus by proposition6.4.9, asEis Mackey as it is defined as a Mackey-dual, we have thatEis bornological.

Corollary 6.4.11. WhenEandF are Mackey and complete, their respective dualE_µ¹ andF_µ¹ are bornological.

Since our Mackey-completion6.2.22preserve the dual, we have that:

E» pEx^1M_µ q¹_µ, F » pxF^1M_µ q¹_µ, that isEandFare the Mackey dual of convenient lcs.

Corollary 6.4.12. In particular, ifEandF are Mackey and Complete EεF

is Mackey, asEεF » pE_µ¹ bβF_µ¹q¹_µ.

6.4.1.3 A chirality between convenient and Complete and Mackey spaces

In the previous section, we showed a dual characterization between bornological spaces, whose strong dual is complete, and complete and Mackey spaces, whose Mackey dual is bornological. We know prove a few more results in order to show that we are indeed in the context of a strong monoidal adjunction of duals.

Proposition 6.4.13. IfEis a convenient space andF is complete and Mackey, then we have natural bijections:

L^oppE,xF^1M_µ q »LpE_µ¹, Fq, which thus leads to and adjunction between

p_q¹_µ:CONV //COMPLMACKEY

and p_qˆ^M_µ :COMPLMACKEY //CONV.

Proof. The functors are is well defined: ifEis convenient theE_µ¹ is complete by proposition6.4.6. Conversely if Fis Mackey and complete,xF^1M_µ is convenient by corollary6.4.11. Then the adjunction follows from the universal property of the Mackey-completion for continuous linear maps (see Proposition6.2.22):

LpxF^1M_µ , Eq „LpF_µ¹, Eq

asEis Mackey-complete, and from proposition4.0.10asEandF_µ¹ are Mackey:

LpF_µ¹, Eq “LpE¹_µ, Fq.

Remark6.4.14. Beware that the terminology is particularly uneasy to handle: positives are interpreted by conve-nient lcs, which are the bornological and Mackey-complete ones, while negatives are interpreted by Mackey and Complete lcs. These MackeyandComplete lcs are Mackey-complete as they are complete, but they are more than that.

Proposition 6.4.15. WhenEand F are complete Mackey spaces, then we have a bounded linear isomorphism (that is an isomorphism inCONV):

pEεF{q¹_µ Proof. IfEandF are complete and Mackey, we have by definition

pEεFq¹_µ“LεpE_γ¹, Fq “LµpE_µ¹, Fq

as equicontinuous inE¹_γ are the absolutely convex compact subsets ofE, which are exactly the weakly compact as E is Mackey. AspEz_µ¹q

M

andpFz_µ¹q

M

are convenient, we have on the other hand that: pEz_µ¹q

Mbˆ^M_β zpF_µ¹q

M

is convenient, thus in particular Mackey and linearly homeomorphic to its double Mackey dual. Thus we get

pEz_µ¹q

via the functoriality of Mackey-Completion6.2.22and Lemma6.4.8.

Theorem 6.4.16. To sum up the two previous propositions, we have a strong monoidal adjunction:

pCONV,bp^M_β q pCOMPLMACKEY^op, εq

p_q¹_µ

p_qz¹_µ^M

%

where the composition of the adjoint functors is the identity onCONV. They satisfy forEandF objects ofCONV

andGPCOMPLMACKEY:

LpEbp^M_β F,Gˆ^Mq »LpE, F_µ¹ `Gq. (6.1) Proof. We are left with proving equation6.1. ConsiderEandFconvenient spaces andGa complete space. Then:

LppEbˆβFq, Gq »LpEbβF, GqasEbβFis bornological (prop.6.3.10

»LpE,LpF, G¹_µqqby the universal property ofbβin BORNVEC

»LpE,LβpF, G¹_µqqasEandF are bornological

»LpE,LµpF, G¹_µqqby Proposition6.4.2

»LpE, F_µ¹ `GqasF andGare endowed with their Mackey-topology

By functoriality of the Mackey-completion, and asF is already Mackey-complete, this is extended to an isomor-phism

LpEbp^M_β F,G¯^Mq »LpE,pF_µ¹ `Gq^Mq.

Remark6.4.17. Wenotare here in presence of apositive chirality, as wedo not havean adjunction:

CONV COMPLMACKEY

p_q˜

p_q¯^born

%

The functors are well defined : the completionE˜of a complete space is till endowed by its Mackey-topology by [44, 8.5.8]. The bornologificationF¯^bornof a complete space (thus in particular Mackey-complete) is still Mackey complete asF¯^bornandFhave the same bounded subsets.

Moreover, by the universal property of the completion one hasLpE, F˜ q “ LpE, FqwhenF is complete. As Eis bornological we haveLpE, Fq “LpE, Fqwhich embeds boundedly inLpE,F¯^bornqasF andF¯^born have the same bounded subsets. However this embedding is not a bijection. Thus the categorical setting developed in Section2.2.2does not apply strictly speaking. It is however not an issue: both CONVand COMPLMACKEYembed fully and faithfully in the category TOPVECand we interpret proofs as arrows in TOPVEC(that is, as plain linear continuous maps). Thus a proof of$N is interpreted as a functionf PLpR,JNKq(and notf PLpKˆ1J,JNKqas axiomatized in Section2.2.2) and a proof of$P,N as an arrowf PLppJPK

1µ,JNKqas usually.

Example 6.4.18. A typical complete and Mackey lcs is any Banach or Fréchet space. In particular, spaces of smooth functionsC⁸pRⁿ,Rqare complete and Mackey as they are metrisable and thus bornological. A typical convenient space include any of the previous example, as well as any space of distribution which could be Mackey-complete without being Mackey-complete.

Outlook8. It may be enough to consider ultrabornological spaces instead of convenient ones as an interpretation for positive connectives.