3.1.1 Topologies on vector spaces
Linear topologies
Definition 3.1.1. [44, 6.7] A topological vector space is aK-vector spaceEoverK“RorK“C, which is also a topological space, making addition and scalar multiplication continuous.
The topology of a lcsE, that is the collection of all its open sets, is denotedTE. A subsetU ĂEis aneighborhood ofxPU if it contains an open set containingU.
Definition 3.1.2. A topological vector space is Hausdorff if its topology separates points: for anyx, y PEsuch thatx‰ythere exists two open setsU, V ofEsuch thatUXV “ H,xPU andyPV.
Definition 3.1.3. A subsetU PEisabsolutely convexif for allx, yPY, allλ, µPKsuch that|λ| ` |µ| ď1we haveλx`µv PY. It is absorbent if for anyxPE there existsλP Ksuch thatxPλV. A topological vector space is said to belocally convexif it admits a sub-basis ofabsolutely convex absorbentopen sets.
Closed and absolutely convex subsets of a topological vector spaceE are also called disksinE. The role of absolutely convex neighborhoods is to allow for linear combinations: if Y is absolutely convex, then for any λ, µPKwe have the following equality between sets:λU`µU “ pλ`µqY.
Notation 3.1.4. IfEis a locally convex topological vector space, we writeVEpxqthe set of neighborhoods ofx inE. IfBis a subset ofE, we writeacxpBqthe absolutely convex hull ofBinE, that is the smallest absolutely convex set containingB. We will writeBfor the closure ofB isE, that is the smallest closed set containingB.
When the topology onEis ambiguous, we will writeBTEfor the closure ofBinEendowed withTE.
AsTEis stable under translations (addition is continuous), it is enough to specify a basis of0-neighborhoods to understand the topology ofE. It will be called a0-basisforE.
Proposition 3.1.5. IfE admits a0-basis of absorbing convex open sets, then the absolutely convex closures of these sets is also a 0-basis forE [44, 2.2.2]. Their closure also forms a 0-basis. As any0-neighborhood is absorbent [44, 2.2.3], we will sometimes consider without any loss of a generality0-basis consisting of closed convex sets.
We sum up all the preceding definitions in the one of lcs, which will be our central object of study:
Terminology 3.1.6. We will denote by lcs a topological vector space which is locally convex and Hausdorff.
Proposition 3.1.7. Any lcs is also a uniform space, whose entourage are the setstx´y |x, yPUu Y ty´x| y, xPUufor every0-neighborhoodU.
Isomorphisms
Notation 3.1.8. We shall writeE „F to denote an algebraic isomorphism between the vector spacesEandF, andE »F to denote a bicontinuous isomorphism between the lcsEandF. We will sometimes refer to the first as linear isomorphism and to the second as linear homeomorphism.
The following proposition is deduced from the stability of the topology under sums and scalar multiplication and captures the intuitions of lcs:
Proposition 3.1.9. A linear function between two lcs is continuous if and only if it continuous at0.
3.1.2 Metrics and semi-norms
From now on and for the rest of the thesis,E,FandGwill denote Hausdorff and locally convex topological vector spaces. Let us describe some specific classes of lcs that the reader may be more familiar with:
• Metrizable spaces are the lcs such that the topology is generated by ametric, that is a positive symmetric subadditive applicationdist:EˆE //R`which separates points. Open sets are then generated by the 0-neighborhoodsUn“ txPE|distp0, xq ănu.
• Normed spaces are the lcs such that the topology is generated by anorm, that is a positive homogeneous1 subadditive applicationk¨kE //R`such thatkxk“0if and only ifx“0. Open sets are then generated by the0-neighborhoodsUn “ txPE|kxkănu. Normed spaces are in particular metrizable spaces.
• Euclidean spaces are finite-dimensionalR-vector spaces. They are normed vector spaces, with all norms equivalent tok¨k2:xPRnÞÑař
ix2i.
Metrizable spaces are also the lcs which admit a countable basis, and that’s what makes them useful:
Proposition 3.1.10. [44, 2.8.1] A lcs is metrizable if and only if it admits a countable basis of0-neighborhoods.
Proposition 3.1.11. [44, 2.9.2] Any lcs is linearly homeomorphic to a dense subset of a product of metrizable lcs.
The notion of continuity on a lcs can be equivalently described in terms of topology or in terms of semi-norm. A semi-normon a vector space is a subadditive homogeneous positive applicationp: E //R`. It is notably not required thatppxq “0implies thatx“0.
Local convergence. The following definition will be used to define precise notions of convergence and com-pleteness. It allows to consider convergence locally, within normed spaces generated by some subsets ofE. A typical example of a semi-norm is theMinkowski gauge, defined on a vector spaceE for an absorbing convex subsetAĂE:
qA:xÞÑinft|λ| |λPR˚`, xPλAu.
The gauge is not a norm asqApxqmay be null forx‰0. Forxin the linear span ofAwe have however thatqA separates the points.
Definition 3.1.12. ConsiderU an absolutely convex subset ofE. Then one defines the normed spaceEU as the linear span ofU, endowed with the normqU.
1that is, the norm must verify:@λą0,kλxk“λkxk
Let us explain the equivalence of the description ofTE in terms of topology or in terms of semi-norms (see for example the introduction of Chapter II.4 in [66] for a more detailed presentation). Consider a familyQof semi-norms onE. Then the topology induced by the semi-normsq P Qis the projective topology generated by this family, that is the smallest topology containing the reverse image of any open set ofR˚` byq PQ. A0-basis for this topology is the collection of all
BA1,...,An,“ txPE| @iqAipxq ău.
Consider converselyEa lcs andU a0-basis consisting of absolutely convex subsets. Then the topologyTEofE coincides with the one generated by the family of semi-normspqUqUPU.
3.1.3 Compact and precompact sets
Definition 3.1.13. A subsetKĂEis:
1. compactif any net inKhas a sub-net which converges inK.
2. precompactif for any0-neighborhoodUthere exists a finite setM PEsuch thatKĂU`M. 3. relatively compactif its closure is compact inE.
Since a continuous function preserves open sets by inverse image, it follows that:
Proposition 3.1.14. Iff :E //F is a continuous function, and ifKis a compact inE, thenfpKqis a compact inF.
The preceding proposition applies in particular toId:Eτ1 //Eτ, whenτ1is finer thanτ:
Corollary 3.1.15. ConsiderEa lcs endowed with a topologyτ1, andτanother topology onEwhich is coarser thanτ12. Then ifKis compact inEτ1,τandτ1coincide onK.
Because compact sets are preserved by direct images of continuous function, they are used to construct topologies on spaces of continuous function. This is one of the essential bricks of this chapter. In euclidean spaces, that isR and its finite products, compact sets are exactly the bounded and closed sets, but this is not the case in general.
3.1.4 Projective and inductive topolgies
We describe first products and coproducts of lcs.
Definition 3.1.16. Consider aI-indexed family of lcspEiqiPI. We defineś
iPIEi as the vector space product overIof theEi, endowed with the coarsest topology onEmaking allpicontinuous.
IfUj is a basis of0-neighborhoods inEj, then the follwoing is a subbasis for the topology onś
iEi: U “ tUi0ˆś
iPI,i‰i0EiuwithUi0 PUi0. Definition 3.1.17. We defineE:“À
iPIEias the algebraic direct sum of the vector spacesEi, endowed with the finest locally convex topology making every injectionIj : Ej Ñ Eis continuous. Remember that the algebraic direct sumEis the subspace ofś
iEiconsisting of elementspxiqhaving finitely many non-zeroxi. IfUiis a0-basis inEi, then a0-basis forÀ
iEi[44, 4.3] is described by all the sets:
U “Ť8 n“1
Àn k“1
Ť
jUj,kwithUj,kPUj,j PJ,kPN. Note that this topology is finer than the topology induced byś
EionÀ Ei.
Proposition 3.1.18([44, 4.3.2]). I is finite if and only if the canonical injection from À
iPIEi toś
iPIEi is surjective.
2That is we have an inclusion between the sets of open-sets:τĎτ1
Proof. By the definition of the algebraic co-product, as the sets of elements of the product with finitely many non-zero composites, we haveÀ
iPIEi “ś
iPIEiwhenIis finite. The description of the sub-basis of the inductive and projective topologies gives us the result. The converse proposition follows from the definition of the co-product as the set of finite tuples of the product: if product and co-product coincide, then the indexIis finite.
The previous constructions are in fact particular casse of projective and inductive topologies, defined on pro-jective and inductive limits of topological vector spaces. On these notions, we refer respectively to [44, 2.6] and [44, 4.5, 4.6] and just recall the main definitions and results below.
Definition 3.1.19. ConsiderpEjqjPJa family of lcs indexed by an ordered setJand a systempTj,k:Ek //Ejqjďk Definition 3.1.20. ConsiderpEjqjPJ a family of lcs indexed by a pre-ordered setJ and a system of continuous linear mapspSk,j:Ej //Ekqjďksuch thatSj,j“IdEj and such that foriďjďkwe haveSi,k “Si,j˝Sj,k. continuous, where Qdenotes the linear continuous projection ‘iEi // lim
ÝÑiEi. This topology is not necessarily Hausdorff even when all theEjare.
• The limit is said to be reduced when the mapsSiare injective, and regular when any bounded3set inlim ÝÑiEi
is bounded in one of theEi.
• A reduced and regular inductive limitlim
ÝÑjEjis hausdorff if and only if all theEjare.
• The limit is called strict whenJ “Nand the mapsSi,j are lcs inclusionsEi ĂEj. IfEiis closed inEj
wheniďj, then the limit is regular.
Remark3.1.21. The fact that a condition on the bounded sets is necessary to guarantee a Hausdorff inductive limit is important. Bounded sets, and inductive limits, are well-behaving with positive connectives.
3.1.5 Completeness
The mere data of a topology is in general not enough to have a rich analytic theory. One needs another tool to deduce convergence of limits and then continuity of functions: we require some notion ofcompletenesson spaces.
Definition 3.1.22. A netpxαqαPΛĂEis said to be aCauchy-netif, for any0-neighborhoodU, there existsαPΛ such that for allβěαwe have:xβ´xαPU. A lcsEis said to becompleteif every Cauchy-net inEconverges.
A normed space which is complete is called a Banach space. A metrizable space wich is complete is called aFréchet space or (F)-space. A closed subset of a complete space is complete. Products and co-products of complete spaces are complete.
Theorem 3.1.23. IfEis any lcs, then there exists a complete lcsE˜ with a linear homeomorphismI :E //E˜ such thatIpEqis dense inE. Then˜ E˜is unique up to linear homeomorphism.
Proof. We sketch the construction of [44, 3.3.3]. IfE is metrizable, there exists an isometric embedding ofE in`8pEq, the Banach space of bounded sequences inE (see Section3.2.3). We writeE˜ the closure ofIpEq in`8pEq. If E is not metrizable, we embed E in the product of the metrizable space generated by each 0-neighborhood of a0-basis (by Proposition3.1.11), and take the closure ofEin product of the completions of these metrizable space.
We recall the following classical result, which is proved in particular in [44, 3.9.1].
Proposition 3.1.24. Considerf : E //F a uniformly continuous map. Then there exists a unique uniformly continuous mapf : ˜E //F˜ which extendsf. In particular, any continuous linear map inE //F whereF is complete admits a unique continuous linear extension inE˜ //F.
3That is, any set which is absorbed by every0-neighborhood, see Section3.4
The notion of completeness defined above is the most famous one and when it is necessary it will be specified as Cauchy-completeness. However, other notions of completeness can be defined. One can endow a lcs with other topologies inferred from its dual, such as the weak or the Mackey-topology, and from that define an alternative of completeness (see Definition3.4.16). Other notions of completeness can also be defined as convergence of Cauchy-nets in some specific subsets ofE: quasi-completeness is the convergence of Cauchy-nets in bounded sets (see Section3.4).
Let us recall a fundamental properties relating completeness and compactness:
Proposition 3.1.25. [44, 3.5.1] ConsiderEany lcs. A subsetK ĂEis precompact if and only if it is relatively compact inE, that is if and only if˜ K¯ is compact inE.˜
Remark 3.1.26. We will later state Grothendieck’ theorem??, which allows to seeE˜ as a certain topology on a bidual of E. This is fundamental in our approach, and allows a model of a linear involutive negation, as it corresponds to the isomorphism:
ÒP » pPKRqKL of a polarized linear logic allowing to interpret the involutive negation.