Linear functions and their topologies

3.3.1 Linear continuous maps

ConsiderEandFtwo lcs. We writeE^˚the set ofalllinear formsE //K. We denoteE¹thedualofE, that is the space of all continuous linear mappings fromEtoK. We writeLpE, Fqthe vector space of all continuous linear functions fromEtoF. We will see that several topologies make these vector spaces topological vector spaces.

Let us describe a few well-known examples:

Example3.3.1. ConsiderpPR^˚`and let us writep<sup>˚</sup>“ <sub>p´1</sub><sup>p</sup> . Then for the spaces of sequences of measurable maps described in Section3.2, one haspL<sub>p</sub>pµqq<sup>1</sup>“L<sub>p˚</sub>pµq, and in particularp`_pq¹“`_p˚.

Example3.3.2. The dual ofc₀, the space of sequences converging to0endowed withk.k₈, is`<sub>1</sub>. Note that by the preceding proposition the dual of`1is`₈.

Proof. ForxP `1, thenevx : y P c0 ÞÑ ř

xnynis well defined by boundedness ofx, and thus continuous for the normk¨k₁onc0. Conversely, considerφPc¹₀. Asφis linear continuous onc0, there isM ą0such that for everyxPc0|φpxq| ăMkxk₈. Let us defineφn“φpeⁿq,eⁿbeing the sequence such thateⁿ_k “0ifn‰kand eⁿ_n“1. Definexn“ _|φ^φn

n| whenφnis non-null, andxn“0ifφnis null. Let us denotex^N “ px1, ..., xN,0, . . .q

the finite sequence whoseN first terms are thexn. As the sequenceφis converging towards0, thenφpx^Nqis well defined for everyN, and|φpx^Nq| ă1. We just proved that for allNPN^˚:

N

ÿ

n“1

|φn| ăM.

We deduce thatpřN

n“1|φn|qN is a monotonous bounded sequence, and it is then converging. ThusφP`1, and as pφnq<sup>1</sup><sub>n</sub>“φwe have the linear homeomorphism betweenc<sup>1</sup><sub>0</sub>and`1.

Example3.3.3. ConsiderEa perfect sequence space as described in example2.2.3. Then one has a linear isomor-phism fromE^KtoE¹, which maps a sequencepαqnqnto the linear continuous form:

α:px_nqnPEÞÑÿ

n

α_nx_n. In particular, one has`pK

“`p˚and`1K

“`₈.

The central theorem of locally convex vector spaces is the Hahn-Banach theorem. We recall here two formulations of this theorem, the analytic one and the geometric one. We omit here some more general formulations that won’t be needed in this thesis.

Theorem 3.3.4. [44, 7.2.1] ConsiderEa locally convex topologicalR-vector space (not necessarily Hausdorff), andF a subspace ofE. Then every continuous linear form onFextends to a continuous linear form onE.

Corollary 3.3.5. A locally convex topological vector space is Hausdorff iff for everyx P E´ t0uthere exists uPE¹such thatupxq ‰0.

Theorem 3.3.6. [44, 7.3.4] ConsiderEa (non necessarily Hausdorff) locally convex topologicalR-vector space E,Aa closed absolutely convex subset ofE, andKa compact subset ofEsuch thatAXK “ H. Then there existsuPE¹such that for allxPA,yPK:

upxq ăαăupyq.

Corollary 3.3.7. ConsiderAa non-empty closed and absolutely convex subset ofE. Then ifx‰A, there exists uPE¹such that for everyyPA,|upyq| ď1and such that|upxq| ą1.

In particular, ifxPEis such that for every`PE<sup>1</sup>,`pxq “0, thenx“0.

3.3.2 Weak properties and dual pairs

Topological properties onE are usually considered with respect to the topologyTE ofE, but they can also be definedweakly.

The weak topology

Definition 3.3.8. The weak topologyσpE, E¹qonEis the topology of uniform convergence on finite subsets of E¹. That is, a basis of0-neighborhoods ofσpE, E¹qis:

W`1,...,`n,“ txPE| |l1pxq| ă, ....,|lnpxq| ău

for`1, ...., `nPE¹andą0. We denote byEworEσthe lcsEendowed with its weak topology. We will show later that this change of topology onEdoes not change the dual ofE(proposition3.3.13).

The weak topology onEis the inductive topology generated by all thelPE¹. This definition means that one can see the specifications ofEwas properties which are verified when elements are precomposed by everylPE¹: Proposition 3.3.9. A sequencepxnqn P E^Nis weakly convergent towardsx P E, that isσpE, E¹q-convergent towardsx, if and only if for everylPE¹the sequencelppxnqqnconverges towardslpxqinR.

One checks easily that this topology makes addition and scalar multiplication continuous, and that the basis of 0-neighborhoods described above consists of convex sets. From the Hahn-Banach theorem3.3.5, it follows thatE is separated:

Proposition 3.3.10. Eendowed withσpE, E¹qis a lcs.

Proof. As a consequence of Hahn-Banach separation theorem [44, 7.2.2.a], we have thatE¹separates the points ofE: ifx, y P E are distinct, then there existsl P E¹ such thatlpxq ‰ lpyq. This makes Eendowed with its weak topology a Hausdorff topological vector space. It is locally convex asW`1,¨¨¨,`n,is convex as soon asU is convex.

The weak* topology. We will heavily make use of another kind of weak topology. The weak* topology onE¹ is the weak topology generated onE¹byEwhen the later is viewed as a dual ofE¹.

Definition 3.3.11. The weak* topology onE¹is the topologyσpE¹, Eqof uniform convergence on finite subsets ofE. A basis of0-neighborhoods ofσpEqis:

Wx₁,...,x_n,ε“ t`PE<sup>1</sup>| |`px1q| ă, ....,|`pxnq| ăεu

forx₁, ...., x_n P E andε ą 0. OnceE¹ is given or computed from a first topology onE, we construct lcsE endowed with its weak* topology and denote itE_w˚¹ orE_σ˚¹ .

As for the weak topology, we have the following:

Proposition 3.3.12. E¹endowed withσpE¹, Eqis a lcs.

When it can be deduced from the context without any ambiguity, we will denoteEσasEendowed with its weak topology and E_σpEq¹ as E¹ endowed with its weak* topology. The weak topology induced by E¹ onE is the coarsest topologyτonEsuch thatpE_τq¹ “E:

Proposition 3.3.13. We have the linear isomorphismspEσq¹„E¹andpE_σpEqq¹ „E.

The well-known demonstration of this proposition uses the following lemma:

Lemma 3.3.14. ConsiderE a vector space andl, l1, ...ln linear forms on E. Thenl lies in the vector space generated by the familyl1, . . . , ln(denotedVectpl1, ...lnq) if and only ifŞn

k“1Kerplkq ĂKerplq.

Proof. Ifl PVectpl₁, ...l_nqthen clearlyŞn

k“1Kerpl_kq Ă Kerplq. Conversely, supposeŞn

k“1Kerpl_kq ĂKerplq.

Without loss of generality, we can suppose the familytl_kufree. Let us show the result by induction onn. Ifn“1, thenKerplq “Kerpl1qas they have the same codimension, and one hasl “ _l^lpx0q

1px₀ql1for any fixedx0 RKerplq.

Consider nowl, l1, ...lnlinear forms onEsuch thatŞn

k“1Kerplkq ĂKerplq. Then by restrictingltoKerplnqwe obtain scalarsλ1, ..., λn´1such that

l_|Kerplnq“

n´1

ÿ

k“1

λkl_k|Kerplq. ThenKerpl_nq ĂKerpl´řn´1

k“1λ_kl_kq, and we have our result by the casen“1.

Proof of Proposition3.3.13. Let us show first thatpEwq¹ „ E¹. As the weak topology onEis coarser than the initial topology onE, we haveE¹Ă pEwq¹. Consider now a continuous linear formlonEw. Then by continuity ofl, and with the description of the weak topology given in Section3.3.3, there existsl1, ..., ln PE¹ and ą0 such that

lpWl₁,...l_n,q Ă tλPK| |λ| ă1u.

By homogeneity, we have Şn

k“1Kerpl_kq Ă Kerplq and the preceding lemma implies that l P E¹. Thus pE_wq¹ „E¹. Both their topology being the weak* topology induced by points ofE, we havepE_wq¹ »E¹.

We can continue to writeE¹ for the dual of a spaceE, regardless whether it may be endowed with its weak topology:pEwq¹ „E¹. Moreover we will writeE_w¹ forpE¹qw. The linear isomorphismE„ pE_w˚¹ q¹can be lifted to a linear homeomorphism whenEis endowed with the weak topologyσpE¹qandE²is endowed with the weak*

topologyσpE¹q.

Proposition 3.3.15. For any lcsEone hasE_σ» pE_σ˚¹ q¹_σ˚.

From Proposition3.3.13, one deduces the following criterion for continuity between weak spaces.

Proposition 3.3.16. ConsiderEandF two open sets. A linear functionf :E //Fwis continuous if and only if for all`PF<sup>1</sup>, we have`˝f PE¹.

Proof. Iff :E_w //F_wis continuous then for` PF<sup>1</sup> we have`˝f P pE_wq¹ “E¹. Conversely, suppose that for all`PF<sup>1</sup>, we have`˝f PE¹. Then the reverse image of an open setW_l1,...ln,byf containsW_{l1˝f,...ln˝f,}, which is always an open set inEas thel_i˝fare continuous, thusf is continuous.

Example3.3.17. Beware that a perfect sequence spaceE as described in Section2.2.3is not endowed with its weak topology: the normal and the weak topologies induced byE^K“E¹(example3.3.1) differ. Indeed, the first one is induced by all the semi-norms:

q_α:xÞÑÿ

n

|α_nx_n| for allαPE¹while the weak topology is induced by all the semi norms:

q_α1,...,α^k :xÞÑsup

i

ÿ

n

αⁱ_nxn

3.3.3 Dual pairs

Concerning topologies, a lot of notions do not depend strictly on the topology of a lcsE, but rather on its dual: the first example is the one of bounded sets (Proposition3.4.10). Thus one can vary the topology ofE, as long as it does not change the dual, the bounded sets stay the same. We have a precise knowledge of the topologies which do not change the dual (see Theorem3.5.3). This is why the concept of dual pair is fundamental. Some notions however depends on the topology: this is the case of completeness, or of compactness.

The process described earlier betweenEandE¹can be generalized to any pair of vector spaces forming a dual pair.

Definition 3.3.18. A dual pair consists of a pair of vector spacesEandF, and of a bilinear formx¨,¨y:EˆF //K which is not degenerate.

In particular, ifE is a lcs, thenpE¹, Eqendowed with the applicationx`, xy “ `pxqis a dual pair. That this linear form is non-degenerate on the left follows by definition, and that it is non degenerate on the right follows from theorem3.3.7: ifxis such that for every`we have`pxq “0, thenx“0.

Then the previous definition for the weak* topology onE¹ can be generalized to any dual pairpE, Fq, and leads to a weak topologyσpF, EqonF. This is done in chapter 8.1 of Jarchow’s textbook [44]. Note that the role of the vector spaces in a dual pair are symmetric, and thus a dual pair also defines a weak topologyσpE, FqonE.

In fact, from any separating sub-vector spaceF ofE^˚ or evenK^E, one can define a dual pairpE, Fq: the applicationpxPE, `PFq ÞÑ xx, `y “lpxqis then bilinear and non-degenerate.

Proposition 3.3.19. The following function is a linear continuous injection ofEintoE²: ιE :

#EãÑE^1˚

xÞÑevx:p`ÞÑ`pxqq (3.4)

This injection and the idea thatE(orE) can be considered as a subspace of˜ E²is fundamental in this thesis.

In particular, note thatpE, E¹qform a dual pair through this injection, withxevx, `y “ evxp`q “`pxq. Then the weak topology onE¹, inherited from the dual pairpE, E¹q, is exactly the same topology as the weak* topology, inherited from the dual pairpE¹, Eq.

More generally, for any dual pairpF, Eq, the non degenerate bilinear application allows for the consideration ofF as a subspace ofE^˚.

Proposition 3.3.20. [44, 8.1.4] Consider E a vector space andF a subspace ofE^˚. Then the bilinear form x¨,¨y:FˆE //Kis non-degenerate if and only ifF is dense inE^˚endowed with its weak* topology.

Proof. Suppose thatF is dense inE^˚, and considerxPEsuch that for every`PF,`pxq “0.F being dense in E^˚equipped with its weak topology we have in particular that for every`PE<sup>˚</sup>,`pxq “0. AsE^˚separates the points ofEwe havex“0.

Conversely, ifFis not dense inE^˚, then by corollary3.3.7for`PE<sup>˚</sup>zF¯there existsxPEsuch that`pxq ‰0 but`<sup>1</sup>pxq “0for all`¹PF. Sincex‰0, it makes the applicationFˆE //Kdegenerate.

Then we have:

Proposition 3.3.21. [44, 8.1.5] Consider pE, Fqa dual pair. F endowed with its weak* topologyσpF, Eqis complete if and only ifF “E^˚.

Proof. F“E^˚is complete asE^˚ »K^E, which is complete as a product of complete spaces. Conversely, ifFis complete it is in particular closed, and thusF »E^˚by the previous proposition.

This results means in particular thatE_w˚¹ is complete if and only if every linear scalar map`PE^˚is continuous.

This almost never happen: then when looking for reflexive space (interpreting classical Linear Logic) which are also complete (interpreting smoothness), we need therefore to look for other topologies on the dual.