6.2.1 Bornological spaces
In Chapter3, we defined bounded sets in a lcs as the sets which were absorbed by open sets (Definition3.4.7).
They define a bornology (Definition3.4.6), which is thus defined from the topology ofE. The converse is also possible: from a bornology one can define a topology.
Definition 6.2.1. ConsiderBEa bornology on a vector spaceE. We define the topologyTpBqas the collection of sets which absorbs all the elements ofB( the so-called bounded sets). That is, a subsetU ĂEis an open set ofE if and only if for everyBPBEwe have a scalarλPKsuch that:
BĂλU.
Note that this topology on E might enrich the bornology onE. The Von-Neumann bornology for TpBEq includesBEbut might be larger.
The theory of vector spaces endowed primarily with bounded sets (and not open sets) was intensively studied by Hogbe-Nlend [40]. They are an analogue to the theory of topological vector spaces. The following definition of BORNcan be found in [53, 52.1] [26, 1.2] or [6].
Notation 6.2.2. We write BORNthe category of vector spaces with bornologies and linear bounded maps, and TOPthe category of vector spaces with (non-necessarily separated locally convex vector) topologies and linear continuous maps.
From an object of BORNone can construct a lcs whose open sets are exactly the one absorbing the bounded sets. The subsets absorbing all the bounded sets are calledbornivorous.
Definition 6.2.3. ConsiderE PBORNwith bornologyBE. Then a subsetU ĂEis said to be bornivorous if for everyB PBEthere is a scalarλPKsuch thatBĂλU.
We defineBornas the functor from TOPto BORNmatching a vector spaceEwith a topology to he same vector space endowed with its (Von-Neuman) bornology, and which maps a linear continuous to itself. It is well defined as linear continuous functions are in particular bounded. Symmetrically, we consider also the functorTopfrom BORNto TOPwich mapsEto the lcsEwith the topology of bornivorous subsets, and which is the identity on linear bounded functions. We check that this functor is well-defined in the following Proposition.
Proposition 6.2.4. A linear bounded map between two vector spacesEandFendowed with respective bornolo-giesBEandBF defines a linear continuous maps betweenEendowed withTpBEqandFendowed withTpBFq.
Proof. Consider a linear bounded map fromEtoFandV a subset ofV wich absorbs every element ofBF. Then one sees immediately that`´1absorbs every element ofBFand thus that`is continuous betweenEendowed with TpBEqandFendowed withTpBFq.
Proposition 6.2.5. We have an adjunction:
BORN TOP
Top
Born
%
Proof. ConsiderE P BORNandF P TOP. Then a linear functionf : E //BornpFqis bounded if and only if it sends every bounded set ofBE in a setB1 Ă F which is absorbed by every open set of the topology of F. Thus considerU1 Ă F an open set of F. Then for any bounded setB ofE, we have a scalarλsuch that fpBq ĂλU1, thusBĂλf´1pUq, thusf is continuous fromToppEq //F. One show likewise that to continuous linear functionsToppEq //Fcoincides bounded linear ones fromE //BornpFq.
Bornological lcs On some lcs, not-only do the open sets absorb the bounded subsets (by definition of bounded-ness in a lcs), but all the subsets that absorb the bounded sets are open sets. This means that the lcsEis invariant under the composition of functorTop˝Born. These spaces are called bornological [44, 13.1], and they have a nice characterization in terms of bounded linear maps.
Proposition 6.2.6. [44, 13.1.1] A lcsEis said to bebornologicalif one of these following equivalent propositions is true:
1. For any other lcsF, any bounded linear mapf :E //Fis continuous, that isLpE, Fq “LpE, Fq, 2. Eis endowed with the topologyTop˝BornpEq,
3. E is the topological inductive limits of the spacesEB (see Definition3.1.12), forB bounded, closed and absolutely convex.
4. Ehas the Mackey* topology of uniform convergence on the weak compact and absolutely convex subspaces ofE1, and any bounded linear formf :E //Kis continuous.
This definition coincides in fact to the one of aβ-bornological, but we won’t make use of bornologicality for other bornologies.
Proof. The equivalence between the first two Propositions follows from the adjunction6.2.5.
If we have (2), then we have indeed that a0-neighbourhood inindEBabsorbs every bounded set ofE, thus is a0-neighbourhood inE. As a0-neighbourhood inEis always one in theEB, we haveE“indEB.
If we have (3), then as the Mackey topology is preserved by inductive limits [44, 11.3.1] and normed spaces are endowed with their Mackey-topology, we have that Eis endowed with its Mackey-topology. Now a linear bounded form onEis in particular linear bounded on eachEB, thus linear continuous onEB. By definition of the inductive topology,f is linear continuous onE.
If we have (4), consider any bounded linear functionf :E //F. Then from any`inF1,`˝fis bounded, thus continuous. Thusf is continuous fromEtoFw˚. Thusf is continuous fromEµtoFby the adjunction4.0.9.
Notation 6.2.7. We denote byBTOPVECthe category of bornological lcs and continuous (equivalently bounded) linear maps between them.
Vector bornologies
Definition 6.2.8. A bornology is said to be a vector bornology if it is stable under addition and scalar multipli-cation. It is said to be convex if it is stable under convex closure. It is said to be separated if the only bounded sub-vector space inBist0u.
Definition 6.2.9. We consider the category BORNVECof vector spaces endowed with a convex separated vector bornology, with linear bounded maps as arrows.
One of the reasons why working with bornologies instead of topologies is not that popular is because the image byTopofE PBORNVECmay not have a separated topology. Counter examples are given in [26, 2.2], where a separation procedure is used afterwards. However we have:
Proposition 6.2.10. IfEis inBORNVECandToppEqis bornological, then it is a lcs.
Proof. The fact that the topologyTpBEqis convex and makes addition and scalar multiplication continuous is immediate. Let us show thatTpBEqis separated: considerxandy two different points inE. Suppose that all open sets containingxcontainy. Then the vector space generated byxandyis absorbed by any open set. Indeed, consider an open setUinTpBEq. As the pointtxuis bounded there is a scalarλsuch thatxPλU. Thus, asyPU by hypothesis, every elementµx`µ1yis inµ2U for some scalarµ2. Thus the vector space generated byxandy is absorbed by every absorbing open set, it is thus bounded by Proposition6.2.6.
Bornologication
Proposition 6.2.11. ConsiderE a lcs. ThenTop˝BornpEqhas the same bounded sets asE (meaningBorn˝ Top˝BornpEq “BornpEq) and is thus bornological.
What makes this Proposition work is the fact that, when the bornology is already defined as a Von-Neumann Bornology of subsets absorbed by open sets, then considering the topology ofallbornivorous subsets won’t change the bornology.
Proof. The bounded sets ofTop˝BornpEqare those which are absorbed by every bornivorous subsets of E.
Thus in particular they are absorbed by the open sets ofE, and thus they are bounded inE. Nowby definition ofTop˝BornpEq, any open set of this lcs absorbs the bounded sets ofE. Thus the bounded sets ofEare also bounded inTop˝BornpEq.
The fact that it is bornological follows directly: we haveTop˝Born˝Top˝BornpEq “Top˝BornpEq, thus Top˝BornpEqsatisfies the second criterion of Proposition6.2.6.
Proposition 6.2.12. IfEis a lcs,Top˝BornpEqis the coarsest bornological lcs topology onEwhich preserves the bounded sets ofE.
Proof. Any bornological topology onE absorbs the bounded sets ofEand thus contains the open sets ofTop˝ BornpEq.
Definition 6.2.13. We denote byEsbornthe lcsTop˝BornpEq, which is called thebornologificationofE.
From the previous Proposition it follows:
Proposition 6.2.14. We have the left polarized closure:
BTOPVEC BTOPVEC U
Top˝Born
%
in whichUdenotes the forgetful functor, which leaves objects and maps unchanged.
One could likewise characterize the vector spacesEof BORNVECin which every set absorbed by the open sets ofToppEqare exactly the bounded ones. These are the sets for which the bounded linear maps onto are exactly those continuous ontoToppEq. See [26, 2.1] or the introduction of [6] for more detail.
Proposition 6.2.15. [44, 13.5] The direct sum of any family of bornological lcs is bornological, as is the quotient of a bornological lcs by a closed subspace, as is the finite product of bornological lcs.
Because bounded sets in a metrizable lcs are generated by closed balls, one shows easily:
Proposition 6.2.16. Metrisable lcs are in particular bornological lcs, and thus they are endowed with their Mackey-topology.
6.2.2 Mackey-completeness
Mackey-complete spaces were defined in Section2.4.3as those lcs in which Mackey-Cauchy nets were convergent.
This definition is equivalent to the following one:
Definition 6.2.17. A lcsEis said to beMackey-completeif for every bounded and absolutely convex subsetBof E, the normed spaceEB is complete. EB coincides with the linear span ofBwith the normpB : xÞÑinftλP K,xλ PBu, see3.1.12.
Thus we have in particular that Mackey-completeness is inherited by closed bounded inclusions, and thus by closed continuous inclusions.
Let us note that this definition allows to use Mackey-completeness on vector spaces endowed with bornologies and not topologies. Indeed Mackey-completeness depends only of the bounded subsets of a lcsE(and thus depends only of its dualE1by Proposition3.4.10), and does not depends directly of the topology ofE.
Proposition 6.2.18. [53, I.2.15] Mackey-completeness is preserved by limits, direct sums, strict inductive limits of sequences of closed embeddings. It is not preserved in general by quotient nor general inductive limits.
Proposition 6.2.19. [53, I.2.15] LetEandF be lcs. IfF is Mackey-complete, then so isLβpE, Fq.
Proof. ConsiderpfγqpγPΓqa Mackey-Cauchy net inBpE, Fq. Each one of the netspfγpxqqγPΓconverges towards fpxq PF due to the Mackey-completeness ofF. The functionfthus defined is bounded andpfγqpγPΓqconverges towardsf. ConsiderpfγqpγPΓqa Mackey-Cauchy net in BpE, Fq: we are given a netpλγ,γ1q Ă Rdecreasing towards0and an equiboundedBinBpE, Fqsuch that
fγ´fγ1 Pλγ,γ1B.
Consider also x P E. As Bptxuq is bounded inF,pfγpxqqγPΓ is also a Mackey-Cauchy net. Besides, F is Mackey-complete, so each of these Mackey-Cauchy nets converges towardsfpxq P F. Let us show thatf is bounded. Indeed, considerba closed bounded set inE, andU a0-neighbourhood inF. AsB is equibounded, there isλPCsuch thatBpbq ĂλU. Considerγ0 PΓsuch that, ifγ, γ1 ěγ0then|λγ,γ1| ăλ. ConsiderµPC such thatfγ0pbq õU. Then for allγ ěγ0,fγpbq ϵU `λU. Thusfpbqis inpλ`µqU¯, thus in3pλ`µqU¯. We proved thatfpbqis a bounded set, and sofis bounded.
Definition 6.2.20. We denote by MCOthe category of Mackey-complete vector spaces with and continuous linear functions between them.
Kriegl and Michor only make use of a Mackey-completion which is functorial on bounded linear maps. This is very restrictive and made impossible to extend the results of [49] to modelize classical Linear Logic. The developments in this Chapter are made possible by the following key proposition:
Theorem 6.2.21. [17, 3.11] The full subcategory MCO Ă TOPVEC of Mackey-complete spaces is a reflective subcategory with the Mackey completion p_M as left adjoint to inclusion .
Lemma 6.2.22. The intersectionEpMof all Mackey-complete spaces containingEand contained in the completion E˜ ofE, is Mackey-complete and called the Mackey-completion ofE.
We defineEM;0“E, and for any ordinalλ, the subspace
EM;λ`1“ Ypxnqně0PMpEM;λqΓptxn, ně0uq ĂE˜
where the union runs over all Mackey-Cauchy sequences MpEM;λqof EM;λ, and the closure is taken in the completion. We also let for any limit ordinalEM;λ “ YµăλEM;µ. Then for any ordinalλ,EM;λ Ă EpM and eventually forλěω1the first uncountable ordinal, we have equality.
Proof. The first statement comes from stability of Mackey-completeness by intersection. It is easy to see that EM;λis a subspace. At stageEM;ω1`1, by uncountable cofinality ofω1any Mackey-Cauchy sequence has to be inEM;λfor someλăω1and thus each term of the union is in someEM;λ`1, thereforeEM;ω1`1“EM;ω1.
Moreover if at someλ,EM;λ`1 “EM;λ, then by definition,EM,λis Mackey-complete (since we add with every sequence its limit that exists in the completion which is Mackey-complete) and then the ordinal sequence is eventually constant. Then, we haveEM,λĄEpM. One shows for anyλthe converse by transfinite induction. For, letpxnqně0is a Mackey-Cauchy sequence inEM;λĂF :“EpM. ConsiderAa closed bounded absolutely convex set inFwithxn //xinFA. Then by [44, Prop 10.2.1],FAis a Banach space, thusΓptxn, ně0uqcomputed in this space is complete and thus compact (sincetxu Y txn, ně0uis compact in the Banach space), thus its image inE˜is compact and thus agrees with the closure computed there. Thus every element ofΓptxn, ně0uqis a limit inEA of a sequence inΓptxn, ně0uq ĂEM;λ thus by Mackey-completeness,Γptxn, ně0uq Ă F.We thus conclude to the successor stepEM;λ`1ĂEpM, the limit step is obvious.
Thus we have a left polarized closure
TOPVEC MCO
p¨M
U
%
We will also make use of Lemma 3.7 of [17], which relates Mackey-completions and Mackey-duals.
Proposition 6.2.23. ConsiderEa space endowed with its Mackey-topology. ThenEˆM is still endowed with the Mackey-topologyµpEˆM, E1q.
Proof. Remember that our Mackey-completion preserves the dual, thusµpEˆM, E1qis indeed the Mackey topology onEˆM. MoroeverEˆM is constructed as the intersectionEpM 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 alsoFˆM, which is in between them. Thus the continuous inclusionsppFµ1q1µq //pFˆMq1µq1µ //ppFqr 1µq1µhave always the induced topology. In the transfinite description of the Mackey completion, the Cauchy sequences and the closures are the same inppFrq1µq1µandFr(since they have same dual hence same bounded sets), therefore one finds the stated topological isomorphism.
6.2.3 Convenient spaces
In this section, we develop the theory of convenient spaces as defined by Frolicher and Kriegl [26], and studied in [6].
Definition 6.2.24. A lcs is said to be convenient if it is Mackey-complete and bornological.
Again, beware that the spaces called convenenient in [53] or [49] are just Mackey-complete, and not bornolog-ical. We defined Mackey-complete spaces as those spaces for whichEB is always a Banach, whenB is an absolutely convex and closed bounded subset ofE(definition6.2.17).
By definition2.4.22, a Mackey-Cauchy net is exactly a net which is Cauchy for one of the normspB. Thus by Proposition6.2.6a Mackey-complete bornological space is an inductive limit of Banach spaces.
Definition 6.2.25. A lcsEis said to beultra-bornologicalif it can be represented as an inductive limit of Banach spaces (see [44, 13.1]).
Ultrabornological are in particular barrelled (see Section3.4.4). Barrelledness is a very strong property, which allows for a Banach-Steinhauss theorem [44, 11.1.3] saying that simple convergence and strong convergence are the same for linear functions.
Definition 6.2.26. A set of functionsB:tf :E //Fubetween two lcs is said to be simply bounded if for every xPE, the settfpxq|f PBuis bounded inF. It is said to be strongly bounded, or equibounded, if for any bounded setB ĂE, the setŤ
fpBqis bounded inF. Simply bounded sets of linear continuous functions are exactly the bounded subsets of LσpE, Fq, while strongly bounded sets of linear continuous functions are exactly bounded subset ofLβpE, Fq.
It is clear that strongly bounded sets of functions are in particular simply bounded. We show that under an assumption of Mackey-completeness, the converse is true for linear bounded functions
Proposition 6.2.27. ConsiderEa Mackey-complete space. Then a subsetBĂLpE, Fqis simply bounded if and only if it is strongly bounded.
Proof. Letbbe a bounded set ofE. Taking the absolutely convex closed closure ofb, and becauseEis Mackey-complete, we can assume without loss of generality thatEBis a Banach space. Since any bounded linear maps on a Banach space is also continuous, we can apply the classical Banach Steinhaus theorem on the restriction ofBto linear bounded functions onEB: it is equicontinuous. In particular, it sends bounded sets ofEB to bounded sets ofF andBpbqis bounded.
Corollary 6.2.28. IfEis Mackey-complete and bornological, then a subsetB ĂLpE, Fqis simply bounded if and only if it is strongly bounded.
A fundamental property of convenient spaces is that Mackey-completeness is preserved by bornologification:
Proposition 6.2.29. IfEis a Mackey-complete lcs, thenEsbornis Mackey-complete.
Proof. By Proposition6.2.11 the lcsEsborn :“ Top˝BornpEqhas the same bounded sets asE. As Mackey-completeness only depends on the bounded sets, ifEis Mackey-complete so isEsborn.
Thus if we Mackey-complete and then bornologify a lcsE we obtain a Mackey-complete and bornological vector space.
Notation 6.2.30. We denote byCONVthe category of bornological Mackey-complete lcs, also calledconvenient spaces, and continuous linear maps between them. These are the convenient spaces in [6] and [26]. We denote by
E¯Conv“EsM,born
the closure operation which makes a lcs convenient, that is the bornologification of the Mackey-completion. Thanks to Theorem6.2.21, it enjoys a universal property with respect to linear bounded (hence continuous) maps fromE to a Mackey-complete lcsF.
In diagrams, this amounts to say that we have a polarised closure:
CONV MCO
U
¯¨born
%