Nuclear spaces

Proof. The adjunction follows from the fact that a (F)-space is endowed with itsγ-topology, or said otherwise that a (F)-space isγ-reflexive. Thus the dualf¹of a linear mapf :E_γ¹ //F is a linear mapf¹:F_γ¹ //pE_γ¹q¹_γ, asE is endowed with its Mackey topology (see [44, 8.6.5]). See section3.5.3

We would like to extend this closure to a model of DiLL. This asks for a strong monoidal functor playing the role of an exponential. However, to prove this strong monoidality the setting of nuclear spaces is better-suited.

The Kernel theorem7.3.9, proving the strong monoidality of an exponential interpreted as a space of distributions, is also valid in our setting as spaces of distributions are Schwartz, but it uses the nuclearity of the spaces of distributions.

7.2.2 Nuclear spaces

The theory of nuclear spaces will allow us to interpreted the involutive negation ofDiLL, and as the same time the theory of exponentials as distributions. Nuclear lcs can be understood through two approaches: one uses the theory of nuclear operators [44, 17.3] [76, 47] while the other uses the theory of topological vector spaces. We refer to the book by Treves [76] for a nice introduction to these notions, although [44] contains all the necessary material also.

Definition 7.2.17. A linear mapf from a lcsEto a BanachXis said to be nuclear if there is an equicontinuous sequencepanqinE¹, a bounded sequencepynqinX, and a sequencepλnq Pl1such that for allxPE:

fpxq “ÿ

n

λnanpxqyn.

Definition 7.2.18. ConsiderEa lcs. We say thatEis nuclear if every continuous linear map of E into any Banach space is nuclear.

Proposition 7.2.19. [76, Thm 50.1] The following propositions are equivalent:

• Eis nuclear,

• For every for every Hausdorff spaceE, there is a linear homeomorphism EbπF »EbF,

• For every semi-normponE, there is a semi-normqfiner thanpsuch that the continuous linear injection pE˜pq¹ãÑ pE˜qq¹is nuclear.

Remember from section3.1.2thatE_p denotes the normed spaceE{Kerppqnormed byp, thatE˜_p is thus a Banach space, andpE˜pq¹endowed with the dual Banach topology.pE˜pq¹is also the linear span inE’ of the polar Up˝of the closed unit ballUpofp.

One of the interests of nuclearity lies in its remarkable stability property:

Proposition 7.2.20([44] 21.2.3). The class of nuclear spaces is preserves by completions, projective limits, count-able inductive limits, projective tensor products, subspaces and quotients.

Proposition 7.2.21. The only nuclear normed spaces are the finite-dimensional ones.

Proof. This uses the fact that bounded sets of nuclear lcs are precompact [76, prop. 50.2]: nuclear maps are in particular compact (sending bounded onto relatively compact subsets). IfEis nuclear, then for every absolutely convex neighbourhoodU ofEthe linear mappingE //E˜U is compact, whereEUdenotes the normed associated withU, that is the quotient ofEby the kernel of the semi-norm associated withU. In particular, the image of a bounded setBis compact in any of theE˜U, thus precompact inE.

If moreoverEis normed, then every bounded set is relatively compact, that isEhas the Heine-Borel property.

But only finite-dimensional normed space have the Heine-Borel property.

This proposition is important for the development of Smooth DiLL in section7.4. It means that we have a gap between finite data (Rⁿ) and the smooth computations we make from these dataC⁸pRⁿq.

Example7.2.22. The following lcs, introduced in3.2, are nuclear (see [76, ch.51] for the proofs):

• Any finite dimensional vector spaceRⁿ,

• The spaces of smooth functionsDpRⁿq “ C_c⁸pRⁿ,Rq,EpRⁿq “ C⁸pRⁿ,Rqand their dualsD¹pRⁿqand E¹pRⁿq(see section3.2).

• The Köthe spacesof rapidly decreasing sequences:

s“ tpλnqnPK^N| @kPN,pλnn^kqnP`1u.

This definition generalizes to spaces of rapidly decreasing sequences of tuples inpR^mq^Z.

• The space of tame functions

SpRⁿq:“ tf PC⁸pRⁿq,@P, QPRrX1, ..., Xns,sup_x|PpxqQpB{Bxqfpxq| ă 8u¹ and its dualS¹pRⁿqthe space of tempered distributions. These spaces will be studied in section7.3.4.

Interestingly, the nuclearity of the considered spaces of functions relies on the nuclearity of the Köthe spaces.

Indeed, the space of tempered functions is isomorphic to a subspace ofs, the space of compact supported smooth functions is a subspace of the space of tempered functions; and the space of smooth functions is the projective limit of the space of smooth functions with compact support.

Outlook10. This fact hints for a general correspondence between smooth models of LL and Köthe spaces, via Fourier transform (see [76, Theorem 51.3]).

Lemma 7.2.23. [66, III.7.2.2] Every bounded subset of a nuclear space is precompact.

1The notation QpB{Bxqwill be used in Chapter 8, and represents a linear partial differential operator with finite coefficients. If QpX1, ...Xnq:“ř

αPNⁿaαX₁^α1...X^αnⁿ, thenQpB{Bxqrepresents the operator fPC⁸pRⁿq ÞÑ

˜ xÞÑÿ

α

aα

B^|α|f Bx^α₁¹...Bx^αnⁿ

pxq

¸ .

Proof. ConsiderEa nuclear space andBbounded inE. ConsiderU a basis of absolutely convex closed neigh-bourhoods ofE. As a lcs,Eembeds in the product of Banachś

UPUE˜U, and thusBembeds intoś

UPUE˜U as a product of bounded subsets via mapsqU :E //EĂU. These maps are in particular nuclear. Now we use the fact that nuclear maps are compact [76, 47.3], that is send a0-neighbourhoodU (the polar of the closure of thepanq in definition7.2.17) into a precompact set (the closed hull of theyk) by definition??. ThusB, absorbed byU, is precompact as its image is precompact, and we can recoverB by reverse image byqU of open sets (thus open sets).

Proposition 7.2.24. A complete nuclear lcsEis semi-β-reflexive.

Proof. By lemma7.2.23, the bounded sets of E are compact, and thusF¹ is endowed with theγ-topology of uniform convergence on absolutely convex compact subsets ofF. By the corollary to the Mackey-Arens theorem 3.5.7, this makesF semi-reflexive.

It follows then immediately that:

Theorem 7.2.25. An (F)-spaceF which is also nuclear is reflexive.

Proof. A semi-β-reflexive metrisable space isβ-reflexive: whenF is metrisableE-equicontinuous sets and E-weakly bounded sets coincides inE¹[44, 8.5.1].

Corollary 7.2.26. A fundamental consequence of the previous lemma is that spaces of smooth functions and spaces distributions with compact support, respectively denoted byEpRⁿqandE¹pRⁿq, defined in section are reflexive.

Another property of nuclear spaces is that they allow for a polarized model of MALL with nuclear (F)-spaces and nuclear (DF)-spaces, as the class of nuclear (DF)-spaces is stable bytensor product.

Proposition 7.2.27. • ConsiderEa lcs which is either an (F)-space or a (DF)-space. ThenEis nuclear if and only ifE¹is nuclear [36, Chap II, 2.1, Thm 7].

• IFEis a complete (DF)-space and ifFis nuclear, thenL_bpE, Fqis nuclear. If moreoverFis an (F)-space or a (DF)-space, thenLbpE, Fq¹is nuclear [36, Chapter II, 2.2, Thm 9, Cor. 3]. As a corollary, the dual of a nuclear (DF)-space is a nuclear (F)-space.

Proposition 7.2.28([36] Chapter II, 2.2, Thm 9). IfEandFare both nuclear (DF)-spaces, then so isEbπF. A central result of the theory of nuclear spaces is the following proposition. It is proved by applying the hypothesis thatEis reflexive and thusE¹is complete and barrelled, and thus applying the hypothesis of [76, 50.4].

Proposition 7.2.29([76] prop. 50.4). ConsiderEa nuclear (F)-space, andF a complete space. ThenEbˆπF » LβpE¹, Fq.

Proposition7.2.29follows from the fact thatEbF is dense inLpE¹, Fq, due to the fact that nuclear spaces satisfy a goodapproximation property(see [44, Chapter 18]). Thus whenF is complete we have an isomorphism between the completed tensor product and the space of linear functions. AsEis an (F)-space and thus barrelled, LεpE¹, Fq »LβpE¹, Fq. The following result specifies Buchwalter equalities7.1.11:

Proposition 7.2.30. [76, 50.7] WhenEandF are nuclear (F)-spaces, then writingbˆ equivalently for the com-pleted projective or comcom-pleted injective tensor product, we have:

pEbFˆ q¹»BpE, Fq »E¹bFˆ ¹.

Proof. Remember thatBpE, Fqrepresents the bilinear continuous scalar functions onEˆF. AsEandF are in particular metrizable, we have thatBpE, Fq “BpE, Fq, the space of separately continuous bilinear maps. Thus BpE, Fq “LpE, F¹q, and via Proposition7.2.29we haveLpE, F¹q »E¹bˆπF¹asEis reflexive. Moreover,bπ

is universal for the continuous bilinear maps, and via the universal property for the completion we have moreover:

pEbFˆ q¹»BpE, Fqwhen the last space is endowed with the topology of uniform convergence on bounded subsets ofEbF. Let us show that the linear isomorphismˆ BpE, Fq “ LpE, F¹qis also a linear homeomorphism when BpE, Fqis endowed with this topology convergence on bounded subsets ofEbFˆ . As open subsets ofLpE, F¹q are generated by theWB_E,B_F^˝whereBEandBF are bounded respectively inEandF, we must then show that:

bounded subsets ofEbFare contained in the absolutely convex closure of tensor productsBEbBF of bounded subsets ofEandFrespectively.

However, we know from proposition7.2.20that asEandF are nuclear, so isEbF. Thus bounded subsets of EbF,E andF are precompact thanks to lemma7.2.23, and thus relatively compact as these spaces are (F)-space and in particular complete. However, (F)-(F)-spaces satisfy Grothendieck problèmes des topologies for compact subsets, as already seen in the proof of7.1.12: a compact subset ofEbˆπFis the convex hull of a tensor of compact sets. This can be proved simply by considering that the elements of a projective tensor product of (F)-space are absolutely convergent series

ÿ

n

λ_nx_nby_n whereř

|λn|ă1andpxnqandpynqare sequences converging to0inEandF respectively [76][45.1, corrolary 2]. Thus bounded subsets ofEbF are contained in the convex balanced hull of tensor productsBEbBF of compacts, thus bounded, subsets ofEandFrespectively, and we haveBpE, Fq »LpE, F¹q.