Translator’s note

This page is a translation into English of the following:

Grothendieck, A. “Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires.” Annales de l’institut Fourier 4 (1952), 73-112. DOI: 10.5802/aif.46.

The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.

Version: a721fea

Introduction

Subject

This article aims to give a summary, without proofs, of the principal results found in my work “Produits tensoriels topologiques et espaces nucléaires,” which will be published in the Memoirs of the Amer. Math. Society (and which I will refer to as [PTT]). The main concern throughout [PTT] was that of being exhaustive, both in terms of studying all the questions raised by the topics covered, as well as trying to state the more difficult results as theorems that were as general as possible. This work was also very dense, and the important simple ideas risked being sometimes hidden behind technical details. This is why this bowdlerised summary is possibly useful in giving a more assimilable outline of the theory. Some extra comments, interesting but not necessary for the general understanding of this summary, as well as some hints for certain proofs, have been placed between stars, \star like this \star.

The importance of topological tensor products shows itself in many different settings:

  1. The notion of the topological tensor product forms the foundations of a simple and general formulation of Fredholm theory, including, alongside the classical case of an integral operator defined by a continuous kernel, many other operators that are defined in the most important functional spaces.1

  2. The many variants of the notion of topological tensor product give rise, by duality, to the definition of many remarkable classes of bilinear forms and linear operators, whose study is only just barely covered in [PTT, chap. I, §4]. In particular, the techniques introduced there, conveniently systematised and exploited, allow us to obtain entirely unexpected results in the theory of linear transformations between the spaces L^1, L^2, and L^\infty, and their topological-vectorial analogues (these results being, as of yet, not definitive, and thus unpublished). I might return to this subject, and restrict myself to explaining, in a rather different way, the systematic work of von Neumann–Schatten on the remarkable classes of compact operators in a Hilbert space [8].

  3. From the point of view of this current work, the most important application of topological tensor products is the theory of nuclear spaces. We explain this theory, generalise it, and make precise the famous “theory of kernels” of L. Schwartz, and further discover new properties, even for the most classical of spaces. Here, the topological tensor calculus is the most simple, since the majority of variants of the notion of topological tensor product coincide, and their properties thus sum. For now, there are not many applications of the general theorems that we obtain to specific theories. The most interesting seems to be a topological-vectorial variant of the “Künneth theorem,” giving the homology of a complex defined as the tensor product of two complexes, a variant which seems useful in topological algebra.

  4. Generally, it seems to me that the notions of topological tensor product are perfect for giving a suggestive and manageable language that would be good to use in many situations in functional analysis, especially since we have theorems (some of which are non-trivial) at our disposition from which we can benefit. I hope that this summary (or, better, [PTT]) will succeed in giving the reader a similar impression, before the publication of the articles promised above.

Terminology and notation

Generally we follow the terminology and notation of [3], apart from the fact that we call the semi-reflexive spaces of [3] reflexive. We only consider, unless otherwise stated, spaces that are locally convex and separated; by “quotient space” of a space E, we mean the quotient of E by a closed vector subspace. The dual of E, written E', is assumed to be, unless otherwise stated, endowed with the strong topology (i.e. the topology of bounded convergence). The dual of E', or the bidual of E, written E'', is assumed to be, unless otherwise stated, endowed with the topology given by uniform convergence on the equicontinuous subsets of E', which induces the original topology on E. We will eventually need to appeal to certain notions defined and studied in [6], most notably that of a (\mathscr{DF})-space. For our purposes here, it will suffice to know that the dual of a (\mathscr{F})-space is a (\mathscr{DF})-space; that every normed space is a (\mathscr{DF})-space; and that the dual of a (\mathscr{DF})-space is an (\mathscr{F})-space.

Let E, F, and G be locally convex spaces. Denote by \mathrm{B}(E,F;G) (resp. \mathscr{B}(E,F;G)) the space of continuous bilinear maps (resp. of separately continuous bilinear maps, i.e. linear with respect to each variable) from E\times F to G. Denote by \mathrm{L}(E;F) the space of continuous linear maps from E to F. Denote by \mathscr{B}_e(E'_s,F'_s) the space of separately continuous bilinear forms on the product of the weak duals E'_s and F'_s of E and F, endowed with the biequicontinuous topology, i.e. the topology given by uniform convergence on the products of an equicontinuous subset of E' with equicontinuous subset of F'. This space is complete if and only if the spaces E and F are complete.

We define a bounded (resp. compact, resp. weakly compact) linear map from E to F to be a linear map from E to F that sends a suitable neighbourhood of 0 to a bounded (resp. relatively compact, resp. relatively weakly compact) subset of F.

For short2, if E is a vector space, we say disk or disked set in E to mean a convex and circled (a.k.a. balanced) subset of E. If E is a locally convex space, and A a bounded disk in E, then we denote by E_A the vector space generated by A, and endowed with the norm \|x\|_A=\inf_{x\in\lambda A}|\lambda|. If A is closed, then the unit ball of E_A is A. If A is complete, then E_A is complete. If V is a disked neighbourhood of 0 in E, then E_V denotes the normed space given by passing to the quotient under the semi-norm \|x\|_V=\inf_{x\in\lambda V}|\lambda|.

Recall that a locally convex space is said to be quasi-complete if its closed bounded subsets are complete, barrelled (resp. quasi-barrelled) if the bounded subsets of its weak dual (resp. of its strong dual) are equicontinuous, and bornological if every set of linear forms on E that are uniformly bounded on every bounded subset is equicontinuous. If E is quasi-complete, then barrelled is equivalent to quasi-barrelled; in any case, bornological implies quasi-barrelled.

1 Topological tensor products

1.1 Generalities on E\hat{\otimes}F

[PTT, chap. 1, §1, no. 1 and no. 3]

The axiomatic definition of the algebraic tensor product E\otimes F of two vector spaces E and F, and of the canonical bilinear map (x,y)\mapsto x\otimes y from E\times F to E\otimes F ([1]) asks only that, for every vector space G, the bilinear maps from E\times F to G correspond bijectively with linear maps f from E\otimes F to G, where the map corresponding to f is given by (x,y)\mapsto f(x\otimes y).

If E and F are locally convex spaces, then we can endow E\otimes F with a locally convex topology such that, for every locally convex space G, the continuous bilinear maps from E\times F to G correspond exactly to continuous linear maps from E\otimes F to G. Further, such a topology is unique.

Then the equicontinuous subsets of \mathrm{B}(E,F;G) correspond exactly to the equicontinuous subsets of \mathrm{L}(E\otimes F;G) as well. Unless otherwise mentioned, E\otimes F is assumed to be endowed with the above topology, called the projective tensor product of the topologies of E and F; endowed with this topology, E\otimes F is called the projective topological tensor product of E and F.

If E and F are normed, then E\otimes F is normable, and we can even find a norm such that, for every normed space G, the above isomorphism between \mathrm{B}(E,F;G) and \mathrm{L}(E\otimes F;G) preserves the natural norms. Further, such a norm is unique. This norm on E\otimes F, denoted by u\mapsto\|u\|, where the norms of E and F are implicit, is the lower bound of the quantities \sum_i\|x_i\|\|y_i\| over all representations of u in the form u=\sum_i x_i\otimes y_i (and this norm has already been considered, in [8]). It is also the gauge of the set \Gamma(U\otimes V), where U (resp. V) is the unit ball in E (resp. F), and where U\otimes V denotes the set of x\otimes y such that x\in U and y\in V (with \Gamma denoting, as per usual, the balanced hull). In the case where E and F are general locally convex spaces, a fundamental system of neighbourhoods of 0 in E\otimes F is obtained by taking the sets \Gamma(U\otimes V), where U (resp. V) runs over a fundamental system of neighbourhoods of 0 in E (resp. F).

We can introduce the completion of E\otimes F, denoted by E\hat{\otimes}F, and called the completed projective tensor product of E and F. If E and F are normed spaces, then E\hat{\otimes}F is a Banach space (with a well-defined norm!). If E and F are metrisable, then E\hat{\otimes}F is of type (\mathscr{F}). We have, by definition, the scholium: if E and F are locally convex spaces, and G is a complete locally convex space, then the continuous bilinear maps from E\times F to G correspond bijectively to the continuous linear maps from E\hat{\otimes}F to G.

This claim still holds true for equicontinuous sets of maps. In particular, the dual of E\hat{\otimes}F is \mathrm{B}(E,F), with a correspondence between the equicontinuous subsets (which already suffices to characterise the induced topology on E\otimes F).

I do not know if, when E and F are of type (\mathscr{F}), this algebraic isomorphism from the dual of E\hat{\otimes}F to \mathrm{B}(E,F) is a topological isomorphism, when we endow \mathrm{B}(E,F) with the topology given by bibounded convergence, i.e. uniform convergence on the products of two bounded sets (“the problem of topologies”). An equivalent question is the following: is every bounded subset of E\hat{\otimes}F contained in the closed disked hull of a set A\hat{\otimes}B, where A (resp. B) is a bounded subset of E (resp. F)?

\star We now give some general tips for calculations involving E\hat{\otimes}F ([PTT, chap. 1, §1, no. 3]). If E=\prod_i E_i and F=\prod_j F_j (as topological-vectorial products), then E\hat{\otimes}F can be identified with \prod_{i,j}E_i\hat{\otimes}F_j. If E=\sum_i E_i (as a topological direct sum), and if F is a normable space, then E\hat{\otimes}F can be identified with the topological direct sum \sum_i(E_i\hat{\otimes}F). This remains true if F is an arbitrary (\mathscr{DF}) space, provided that I is countable, and these statements can also be generalised to the case where E is the inductive limit (in the most general sense) of a family E_i of spaces. If E and F are both of type (\mathscr{F}) (resp. type (\mathscr{DF})), then so too is E\hat{\otimes}F. Similarly, if E and F are quasi-normable spaces, or Schwartz spaces (see definitions in [6, Section 3]), then so too is E\hat{\otimes}F. \star

1.2 The space E\hat{\otimes}F when E and F are of type (\mathscr{F})

[PTT, chap. 1, §2, no. 1]

Let E and F be (\mathscr{F}) spaces. Then every element of E\hat{\otimes}F is the sum of an absolutely convergent series of the form u = \sum_i \lambda_i x_i \otimes y_i where (x_i) (resp. (y_i)) is a bounded sequence in E (resp. F), and where (\lambda_i) is a summable sequence of scalars.

(It is also true that, if (x_i), (y_i), and (\lambda_i) are given as above, then the series \sum_i\lambda_i x_i\otimes y_i is always absolutely convergent in E\hat{\otimes}F, and so we have a characterisation of the elements of E\hat{\otimes}F). If E and F are normed, then we can suppose in the above that \|x_i\|\leqslant 1, \|y_i\|\leqslant 1, \sum_i|\lambda_i|\leqslant\|u\|_1+\varepsilon, where \varepsilon>0 is arbitrary and given in advance. In these two statements, if u runs over a compact subset of E\hat{\otimes}F, then we can suppose that the sequences (x_i) and (y_i) remain fixed (and we can even suppose that they are sequences that tend to 0), and that (\lambda_i) runs over a compact subset of \ell^1 (the space of summable sequences). We have an analogous statement for the concrete representation of convergent sequences in E\hat{\otimes}F. Theorem 2 and its previous variants serve mainly as a way to:

Let E and F be spaces of type (\mathscr{F}). Then every compact subset K of E\hat{\otimes}F is contained inside the canonical image of the unit ball of a space E_A\hat{\otimes}F_B, where A (resp. B) is a compact disked subset of E (resp. F). A fortiori, K is contained in the closed convex hull of A\otimes B.

This fact also implies that, on \mathrm{B}(E,F), the “bicompact convergence” topology is identical to the compact convergence topology on the dual of E\hat{\otimes}F.

\star For the proof of Theorem 2, we suppose, for simplicity, that E and F are Banach spaces, and let I be the product of their unit balls, and i\mapsto x_i and i\mapsto y_i the projections from I to its factors. It is easy t see that the linear map (\lambda_i)\mapsto\sum_i\lambda_i x_i\otimes y_i from \ell^1(I) to E\hat{\otimes}F is a metric homomorphism from the former to a dense subspace of the latter, and thus onto the latter, whence it indeed follows that E\hat{\otimes}F can be identified with a quotient space of \ell^1(I).

From Theorem 2, we can extract results of the following type: let {\mathscr{G}} be a locally compact group (resp. a Lie group), then every summable function f (resp. every infinitely differentiable function of compact support) on {\mathscr{G}} is of the form \sum\lambda_i g_i*h_i, where (\lambda_i)\in\ell^1, and where (g_i) and (h_i) are bounded sequences in \mathrm{L}^i({\mathscr{G}}) (resp. in {\mathscr{D}}({\mathscr{G}}), the space of infinitely differentiable functions of compact support on {\mathscr{G}}); we thus immediately conclude that f is a linear combination of functions of positive type that are in \mathrm{L}({\mathscr{G}}) (resp. {\mathscr{D}}({\mathscr{G}})). In the former case, we can also restrict to functions that are all of compact support (with the supports of g_i and h_i being contained inside a compact subset that depends only on the compact support of f). There is a simple direct proof in the case of \mathrm{L}^1({\mathscr{G}}), but I do not think that there is one for {\mathscr{D}}({\mathscr{G}}), where the question presents difficulties even for {\mathscr{G}}=\mathbb{R} (by using the Fourier transformation). \star

1.3 Calculation of \mathrm{L}^1\hat{\otimes}E

[PTT, chap. 1, §2, no. 2]

Let M be a locally compact space endowed with a measure \mu\geqslant 0, E a Banach space, and \mathrm{L}_E^1(\mu) the space of \mu-integrable maps from M to E ([2]) endowed with its usual norm: \|f\|_1=\int\|f(t)\|\mathrm{d}\mu(t). We denote by \mathrm{L}^1(\mu) the space of \mu-summable scalar functions. Then there exists an obvious bilinear map (\varphi,a)\mapsto\varphi\cdot a from \mathrm{L}^1(\mu)\times E to \mathrm{L}_E^1(\mu) that is of norm \leqslant 1, and thus defines a linear map of norm \leqslant 1 from \mathrm{L}^1(\mu)\hat{\otimes}E to \mathrm{L}_E^1(\mu).

The above map from \mathrm{L}^1(\mu)\hat{\otimes}E to \mathrm{L}_E^1(\mu) is a metric isomorphism from the former space to the latter.

To see this, we can immediately reduce to the case where E is of finite dimension, and then proceed by transposition. It then suffices to apply the classical theorem of Dunford–Pettis that characterises continuous linear maps from \mathrm{L}^1(\mu) to E'.

If E is an arbitrary locally convex space, then we denote by \mathrm{L}_E^1(\mu) the completion of the separated space associated to the space of continuous maps with compact support from M to E, endowed with its family of semi-norms f\mapsto\int p(f(t))\mathrm{d}\mu(t) (where p runs over a fundamental family of continuous semi-norms on E). Then Theorem 3 easily implies that \mathrm{L}_E^1(\mu) is again isomorphic to \mathrm{L}^1(\mu)\hat{\otimes}E.

If E is a closed vector subspace of the Banach space F, then the canonical linear map from \mathrm{L}^1(\mu)\hat{\otimes}E to \mathrm{L}^1(\mu)\hat{\otimes}F is a metric isomorphism.

This recovers, for example, the well-known fact that every continuous linear map from E to the dual \mathrm{L}^\infty(\mu) of \mathrm{L}^1(\mu) can be extended to a linear map of the same norm from F to \mathrm{L}^\infty(\mu); or, dually, that every continuous linear map from \mathrm{L}^1(\mu) to a quotient space F'/E^0 of a Banach dual by a weakly closed vector subspace comes from a linear map of the same norm from \mathrm{L}^1(\mu) to F'.

\star The analogue of Theorem 3 for \mathrm{L}^p spaces is false for all p>1. However, Theorem 3 applies, in an essential way, in many important places in the theory outlined here. We now give some less important applications (see [PTT, chap. 1, §2, no. 2] for details). In Theorem 3, if we take E=c_0 (the space of scalar sequences that converge to 0), and note that \mathrm{L}_E^1(\mu) can then be identified with the space of latticially bounded sequences in \mathrm{L}^1(\mu) that converge to 0 almost everywhere, then we see that such sequences in \mathrm{L}^1(\mu) form a category of invariant sequences from the topological-vectorial point of view. In particular, a continuous linear map from \mathrm{L}^1(\mu) to a space \mathrm{L}^1(\nu) sends latticially bounded sequences that converge to 0 almost everywhere to sequences of the same type. We thus also see that the latticially bounded subsets of \mathrm{L}^1(\mu) form an invariant category from the topological-vectorial point of view. If we take F=\ell^p, with 1\leqslant p<+\infty, then we similarly obtain a topological-vectorial interpretation of the sequences (f_i) in \mathrm{L}^1(\mu) such that \int \left( \sum_i |f_i(t)|^p \right)^{1/p} \mathrm{d}\mu(t) < +\infty. Such sequences are sent to sequences of the same type by any continuous linear map from \mathrm{L}^1(\mu) to a space \mathrm{L}^1(\nu). Another interesting application of Theorem 3 is the following: every bounded subset M of \mathrm{L}^1(\mu)\hat{\otimes}E is contained in the canonical image of the unit ball of a space \mathrm{L}^1(\mu)\hat{\otimes}E_A, where A is a closed bounded disk in the space E of type (\mathscr{F}); a fortiori, M is contained in the closed disked hull of B\otimes A, where B is the unit ball of \mathrm{L}^1(\mu), which solves the “problem of topologies” described in §1.1. \star

1.4 Other examples

If H is a Hilbert space, then the elements of H'\hat{\otimes}H, identified with endomorphisms of H (the Fredholm maps or nuclear maps from H to H — see §1.7) are exactly the endomorphisms u such that the positive Hermitian operator \sqrt{u^*u} is compact and has a summable sequence of eigenvalues, and \|u\|_1 is then equal to the sum of the eigenvalues of \sqrt{u^*u} (repeated with multiplicities, of course). We obtain the operators that have already been studied in [4] and [8]. In fact, u is also a Fredholm operator if and only if its Hermitian components \frac12(u+u^*) and \frac{1}{2i}(u-u^*) are, i.e. if they are compact Hermitian operators whose sequences of eigenvalues are summable. Relation to Hilbert–Schmidt operators: If A and B are Hilbert–Schmidt operators, then AB is a Fredholm operator, and \|AB\|_1\leqslant\|A\|_2\|B\|_2; and, conversely, every Fredholm operator u is the product of two Hilbert–Schmidt operators A and B of norm \|A\|_2=\|B\|_2=\sqrt{\|u\|_1}. All of these facts are elementary (one we know the spectral decomposition of compact Hermitian operators to a Hilbert space) and well known.

Numerous other examples of products E\hat{\otimes}F, relating to nuclear spaces, will be seen in §2.5.

\star In the setting of infinite-dimensional Banach spaces, I do now know, even in particular cases, other concrete characterisations of the elements of E\hat{\otimes}F apart from those that we have given. Also, the elements of c_0\hat{\otimes}E (where E can be any complete locally convex space) can be identified with certain sequences in E that converge to 0, that we can call nuclearly convergent to 0; but if E is an infinite-dimensional Banach space, then we always obtain a strictly smaller class than the class of all sequences that converge to 0 (see §2.2, Theorem 2). We even show that (if E is an infinite-dimensional Banach space), for any sequence (\lambda_i) of positive scalars that is not square-summable, there exists a sequence (x_i) in E that does not converge nuclearly to 0, and such that \|x_i\|=\lambda_i for all i. However, if, for example, E is the space C(K) of continuous functions on a compact space, then we show that every square-summable sequence in C(K) converges nuclearly to 0. We also point out that, in any complete locally convex space, every summable sequence converges nuclearly to 0. \star

1.5 E\hat{\hat{\otimes}}F spaces

[PTT, chap. 1, §3, no. 3]

If E and F are Banach spaces, then E\otimes F can be considered as a vector subspace of the Banach space \mathrm{B}(E',F') of continuous bilinear forms on E'\times F'. The completion of E\otimes F under the norm induced by \mathrm{B}(E',F') is denoted by E\hat{\hat{\otimes}}F, which is thus a complete normed vector subspace of \mathrm{B}(E',F'). Every reasonable normed topology on E\otimes F is included between the topology induced by E\hat{\otimes}F and the topology induced by E\hat{\hat{\otimes}}F. If E and F are now both arbitrary locally convex spaces, then we can again consider E\otimes F is a space of bilinear forms on E'\times F', and endow it with the biequicontinuous-convergence topology (i.e. the topology given by uniform convergence on the products of an equicontinuous subset of E' with an equicontinuous subset of F'): the completion of E\otimes F under this topology is again denoted by E\hat{\hat{\otimes}}F. If E and F are complete, then the space \mathscr{L}_e(E'_s,F'_s) of separately continuous bilinear forms on the product E'_s\times F'_s of the weak duals E'_s and F'_s endowed with the biequicontinuous-convergence topology is complete, and so E\hat{\hat{\otimes}}F can be identified with a topological vector subspace of \mathscr{L}_e(E'_s,F'_s). Then the elements of E\hat{\hat{\otimes}}F can be identified with certain separately weakly continuous bilinear maps on E'\times F', or even with certain weakly continuous linear maps from E' to F; these linear maps send equicontinuous subsets of E' to relatively compact subsets of F, and the converse is true in all known cases (see §1, Appendix 2).

The topology on E\otimes F induced by E\hat{\otimes}F is finer than that induced by E\hat{\hat{\otimes}}F, whence we have a canonical continuous linear map E\hat{\otimes}F \to E\hat{\hat{\otimes}}F.

An important, unsolved, problem is the question of whether or not this map is always bijective (see §1, Appendix 2). We note that it seems extremely plausible that, if E and F are Banach spaces such that the above map E\hat{\otimes}F\to E\hat{\hat{\otimes}}F is a topological isomorphism (or even simply a topological homomorphism, i.e. here a map from the first space onto the second), then E or F is of finite dimension. This is true if, for example, E contains a vector subspace isomorphic to an \ell^p space or to c_0.

Let E and F be Banach spaces. The norm induced by the dual of E\hat{\otimes}F on E'\otimes F' is clearly the norm induced by E'\hat{\hat{\otimes}}F'. On the other hand, the norm induced on E'\otimes F' by the dual of E\hat{\hat{\otimes}}F is, in all known cases (see §1, Appendix 2), identical to the norm induced by E'\hat{\otimes}F'. This duality, which barely appears in the present summary, is a precious tool for various questions (e.g. [PTT, chap. 1, §4, no. 6]). The following theorem (which is, truth be told, trivial), can be seen as the dual response to Theorem 3.

Let M be a locally compact space, let \mathrm{C}_0(M) the space of continuous scalar functions on M that are “zero at infinity,” endowed with the uniform convergence norm, and let E be a locally convex complete space. Then \mathrm{C}_0(M)\hat{\hat{\otimes}}E is canonically isomorphic to the space \mathrm{C}_0(M,E) of continuous maps from M to E that are zero at infinity, endowed with the uniform convergence topology (and with the uniform norm if E is a Banach space). In particular, c_0\hat{\hat{\otimes}}E is isomorphic to the space of sequences in E that tend to 0.

As with the spaces C(K)\hat{\otimes}E below, here the spaces \mathrm{L}'(\mu)\hat{\hat{\otimes}}E do not, in general, have a special interpretation as functional spaces. We note, however, that the space \ell^1\hat{\hat{\otimes}}E can be understood as the space of summable sequences in E (i.e. of “commutatively convergent” sequences in E).

1.6 Tensor product of linear maps

[PTT, chap. 1, §1, no. 2]

Let E_i and F_i (for i=1,2) be locally convex spaces, and let u_i be a continuous linear map from E_i to F_i. In algebra, we define a linear map u_1\otimes u_2 from E_1\otimes E_2 to F_1\otimes F_2 by the formula (u_1\otimes u_2)(x_1\otimes x_2)=u_1x_1\otimes u_2x_2. This map is continuous if we endow E_1\otimes E_2 and F_1\otimes F_2 with the topologies induced by E_1\hat{\otimes}E_2 and F_1\hat{\otimes}F_2 (resp. by E_1\hat{\hat{\otimes}}E_2 and F_1\hat{\hat{\otimes}}F_2). It then follows that u_1\otimes u_2 extends, by continuity, to a continuous linear map u_1\hat{\otimes}u_2 from E_1\hat{\otimes}E_2 to F_1\hat{\otimes}F_2 (resp. u_1\hat{\hat{\otimes}}u_2 from E_1\hat{\hat{\otimes}}E_2 to F_1\hat{\hat{\otimes}}F_2). These maps will also simply be denoted by u_1\otimes u_2 whenever there is no fear of confusion.

If each u_i is a topological homomorphism from E_i to a dense subspace of F_i, then u_1\hat{\otimes}u_2 is a topological homomorphism from E_1\hat{\otimes}E_2 to a dense subspace of F_1\hat{\otimes}F_2. If each u_i is a topological isomorphism from E_i to F_i, then u_1\hat{\hat{\otimes}}u_2 is a topological isomorphism from E_1\hat{\hat{\otimes}}E_2 to F_1\hat{\hat{\otimes}}F_2.

These claims remain true if the E_i and F_i are normed and we consider metric homomorphisms and metric isomorphisms. Particularly interesting is the following corollary, which can also be obtained by an application of Theorem 2.

If the E_i and F_i are (\mathscr{F})-spaces (for i=1,2), and the u_i are homomorphisms from E_i onto F_i, then u_1\hat{\otimes}u_2 is a homomorphism from E_1\hat{\otimes}E_2 onto F_1\hat{\otimes}F_2.

As particular cases of this corollary, we obtain interesting lifting properties of vector functions with values in a quotient space of an (\mathscr{F})-space E. If e.g. f is an infinitely differentiable map from an open subset of \mathbb{R}^n to E/F, then it comes from an infinitely differentiable map from this same open subset to E. There is an analogous result for holomorphic functions, or infinitely differentiable functions on \mathbb{R}^n that decay rapidly, or summable functions of a certain measure, etc.3 Another application is the following: let D be a differential operator in the space \mathscr{E}(U) of infinitely differentiable functions on an open subset U of \mathbb{R}^n, let E be an (\mathscr{F})-space, and let \mathscr{E}(U,E) be the space of infinitely differentiable maps from U to E. Then we have \mathscr{E}(U,E)=\mathscr{E}(U)\hat{\otimes}E=\mathscr{E}(U)\hat{\hat{\otimes}}E (see §2.5). Let D_E be the operator in \mathscr{E}(U,E) defined by D, and then D_E=D\hat{\otimes}1, where 1 is the identity on E. Then, if D is a topological homomorphism (resp. an onto topological homomorphism), then so too is D_E. Indeed, in the case of an onto homomorphism, this is a particular case of the corollary of Theorem 5, and, in the general case, we use Theorem 5 as well as the fact that, for every quotient space F of \mathscr{E}(U), we have F\hat{\otimes}E=F\hat{\hat{\otimes}}E (since \mathscr{E}(U), and thus F, are nuclear; see Definition 1 in §2.2 and Theorem 3 in §2.3).

We note that, if u_1 and u_2 are topological isomorphisms, then u_1\hat{\otimes}u_2 is not, in general, a topological isomorphism (nor is, in general u_1\hat{\hat{\otimes}}u_2 a topological homomorphism, if u_1 and u_2 are onto topological homomorphisms). If each E_i is identified with a topological-vectorial subspace of F_i by u_i, then the canonical map u_1\hat{\otimes}u_2 from E_1\hat{\otimes}E_2 to F_1\hat{\otimes}F_2 is a topological isomorphism if and only if every equicontinuous set of bilinear forms on E_1\times E_2 is the set of restrictions of an equicontinuous set of bilinear forms on F_1\times F_2. If F_1 and F_2 are of type (\mathscr{F}), then it suffices to consider sets consisting of one single bilinear form. In general, this criterion will not hold true, but it is linked to an existence problem of topological complements.

\star More precisely, if E_1 and E_2 are direct factors, then E_1\hat{\otimes}E_2 can be identified with a topological-vectorial subspace of F_1\hat{\otimes}F_2. Also, if E is a topological-vectorial subspace of the Banach space F, and if the canonical map E\hat{\otimes}G\to F\hat{\otimes}G is a topological isomorphism when G=F', then E'' is a direct factor of F''; thus, in the frequent case where E is already a direct factor of E'', E will also be a direct factor of F.

A useful case where the tensor product u\hat{\otimes}v of two topological isomorphisms is a topological isomorphism is the following: If E'' is the bidual of E, then E\hat{\otimes}F can be identified with a topological-vectorial subspace (resp. a normed vector subspace, if E and F are normed) of E''\hat{\otimes}F. \star

1.7 Nuclear maps

[PTT, chap. 1, §3, no. 2]

Let E and F be Banach spaces. The continuous bilinear map (x',y)\mapsto x'\otimes y from E'\times F to \mathrm{L}(E,F) defines a natural continuous linear map from E'\hat{\otimes}F to \mathrm{L}(E,F); the elements of the image of E'\hat{\otimes}F in \mathrm{L}(E,F) are said to be nuclear maps from E to F. (We also define, between arbitrary locally convex spaces, the notions of trace maps and Fredholm maps — see Appendix 1 — which coincide with the notion of nuclear maps if E and F are Banach spaces; in this case, we can thus freely switch between saying “nuclear maps,” “trace maps,” or “Fredholm maps.”) Nuclear maps from E to F form a vector space which can be identified with a quotient space of E'\hat{\otimes}F (and with E'\hat{\otimes}F itself in all known cases — see “problem of bijectivity” in §1). The quotient norm, again denoted by u\mapsto\|u\|_1, is called the trace norm of the nuclear operator u.

If E and F are arbitrary locally convex spaces, then a linear map u from E to F is said to be nuclear if it is the composition of a sequence of three operators E \xrightarrow{\alpha} E_1 \xrightarrow{\beta} F_1 \xrightarrow{\gamma} F where E_1 and F_1 are Banach spaces, \beta is a nuclear map from E_1 to F_1, and \alpha and \gamma are continuous linear maps. It is equivalent to say that there exists an weakly closed, equicontinuous, disked subset A of E', and a bounded disked subset B of F such that F_B is complete, and finally some u_0\in E'_A\hat{\otimes}F_B such that u is the operator from E to F defined by u_0 (a priori, u_0 defines a nuclear map from \widehat{E}_{A^0} to F_B). A nuclear map is always compact (i.e. sends a suitable neighbourhood of 0 to a relatively compact set). Even better: by the corollary of Theorem 2, we can suppose (in the above) that B and A are compact subsets of F and strong E' (respectively). Theorem 2, applied directly, also gives the following: nuclear maps from E to F are exactly the maps that are sums of series (always absolutely convergent in \mathrm{L}(E,F) endowed with the bounded-convergence topology) u=\sum\lambda_i x'_i\otimes y_i, where (x'_i) is an equicontinuous sequence in E', (y_i) is a sequence contained in a compact disk of F, and (\lambda_i) is a summable sequence of scalars. If we compose a nuclear map, on the left or on the right, with a continuous linear map, then we obtain another nuclear map. The dual of a nuclear map from E to F is a nuclear map from strong F' to strong E' (and even from F' endowed with the uniform convergence on compact disks of F, to strong E').

Nuclear maps from E to itself (more precisely, the slightly larger category of Fredholm maps from E to itself) form a natural domain for Fredholm theory. Here, our interest lies in other properties of these operators, that result directly from either Theorem 2 or the corollary of Theorem 5.

Let E and G be locally convex spaces, and F a vector subspace of E. Then:

  1. Every nuclear map from F to G is the restriction of a nuclear map from E to G.

  2. Suppose that F is closed, and that every compact disc of E/F is contained in the canonical image of a bounded disc A of E such that E_A is complete (for example, a complete bounded disc). (It suffices, for example, for E to be an (\mathscr{F})-space, or for it be the dual of an (\mathscr{F})-space and for F to be weakly closed.) Then every nuclear map from G to E/F can be obtained from a nuclear map from G to E by passing to the quotient.

We have analogous statements for “equinuclear” sets of maps, if by that we mean a set of maps from a locally convex space M to another locally convex space N, contained in the set of maps defined by the unit ball of a space M'_A\hat{\otimes}N_B, where A is a weakly closed equicontinuous disk of M', and B a bounded disk in N such that N_B is complete.

We note that, if u is a nuclear map from E to F, M a closed vector subspace of E contained in the kernel of u, and N a closed vector subspace of F containing u(E), then, in general, the linear map from E/M to F or from E to N defined by u is not nuclear, even if E and F are reflexive Banach spaces. However, if M (resp. N) admits a topological complement, then the map from E/M to F (resp. from E to N) defined by u will itself be nuclear. This is the case, in particular, if E (resp. F) is a Hilbert space.

1.8 Integral linear maps, integral bilinear forms

[PTT, chap. 1, §4, no. 3 and no. 4]

Let E and F be locally convex (resp. normed) spaces, and v a separately continuous bilinear form on E\times F. Then the following conditions are equivalent:

  1. u\mapsto\langle u,v\rangle is a linear form on E\otimes F that is continuous for the topology induced by E\hat{\hat{\otimes}}F (resp. u\mapsto\langle u,v\rangle is a linear form on E\otimes F of norm \leqslant 1 when E\otimes F is endowed with the norm induced by E\hat{\hat{\otimes}}F).

  2. v is contained in the closed disked hull in \mathscr{B}_\mathrm{s}(E,F) (the space \mathscr{B}(E,F) endowed with the simple-convergence topology) of a set M\otimes N, where M is an equicontinuous subset of E', and N an equicontinuous subset of F' (resp. M the unit ball of E', and N the unit ball of F').

  3. There exists a measure \mu on the product space of a weakly compact equicontinuous subspace M of E' with a weakly compact equicontinuous subspace N of F' (resp. a measure \mu of norm \leqslant 1 on the product of the unit ball of E' with the unit ball of F', endowed with the product of the weak topologies) such that we have the formula v = \int_{M\times N} x'\otimes y' \mathrm{d}\mu(x',y') (the weak integral in \mathscr{B}(E,F), in duality with E\otimes F).

  4. There exists a compact space endowed with a positive measure \mu of norm \leqslant 1, a continuous linear map \alpha from E to \mathrm{L}^\infty(\mu), and a continuous linear map \beta from F to \mathrm{L}^\infty(\mu) (resp. the same, but with \alpha and \beta also of norm \leqslant 1) such that we have u(x,y)=\langle\alpha x,\beta y\rangle for x\in E, y\in F.

A bilinear form on E\times F is said to be integral if it satisfies any of the equivalent conditions of Theorem 7. In particular, the dual of E\hat{\hat{\otimes}}F can be identified with the space \mathrm{J}(E,F) of integral bilinear forms on E\times F. If E and F are Banach spaces, then \mathrm{J}(E,F) will be endowed with the dual norm of the Banach space E\hat{\hat{\otimes}}F, which we call the integral norm, and denote by \|v\|'_1. Similarly, a linear map v from one locally convex space E to another G is said to be integral if the corresponding bilinear form on E\times G' is integral.

If E and G are Banach spaces, then we also call the integral norm of the corresponding bilinear form of v the integral norm of v.

Recall that, in all known cases, when E and F are Banach, the natural linear map from E'\hat{\otimes}F' to \mathrm{J}(E,F) is a metric isomorphism from the first space to the second (see §1.5), which is why we use the notation \|v\|'_1 for the integral norm, closely related to the notation \|v\|_1 for the trace norm. Criterion (d) of Theorem 7 takes the following form (which we state for Banach spaces, for simplicity) for integral linear maps: Let v be a linear map from one Banach space E to another F. Then v is integral and of integral norm \leqslant 1 if and only if the map from E to F'' that it defines can be obtained by composing a linear map of norm \leqslant from E to some space \mathrm{L}^\infty(\mu), constructed with a suitable positive measure of norm \leqslant 1 on a compact space, with the identity map from \mathrm{L}^\infty(\mu) to \mathrm{L}^1(\mu), and finally with a linear map of norm \leqslant from \mathrm{L}^1(\mu) to F''. Similarly, criterion (b) of Theorem 7 easily gives: The linear map v from E to F is integral and of integral norm \leqslant 1 if and only if it is an adherent point in \mathrm{L}_s(E,F_s) (where F_s denotes F endowed with the weak topology, and \mathrm{L}_s(E,F_s) denotes \mathrm{L}(E,F_s) endowed with the simple-convergence topology) of the disked hull of the set of the x'\otimes y, where x' (resp. y) runs over the unit ball of E' (resp. of F); or if it is adherent to the set of nuclear operators of trace-norm \leqslant 1.

Let E and F be arbitrary locally convex spaces. Then every bilinear form on E\times F defined by an element of a space E'_A\hat{\otimes}F'_B, where A (resp. B) is a weakly closed disked equicontinuous subset of E' (resp. of F'), is integral. Thus every nuclear map from E to F is integral. The converse is false, even for Banach spaces, since the identity map from \mathrm{L}^\infty(\mu) to \mathrm{L}^1(\mu) is integral, but it is not, in general, even compact. If E=\mathrm{C}(M) (resp. F=\mathrm{C}(N)) is the space of continuous scalar functions on the compact space M (resp. N), then we have seen (Theorem 4) that \mathrm{C}(M)\hat{\otimes}\mathrm{C}(N) can be identified with its norm to the space \mathrm{C}(M\times N), and so the space of integral bilinear forms on \mathrm{C}(M)\times\mathrm{C}(N) can be identified with its norm to the space of Radon measures on the compact space M\times N. Other examples will be seen in §1.9.

By composing an integral linear map on the left or on the right with a continuous linear map, we obtain another integral linear map. The transpose of an integral map from E to F is an integral map from strong F' to strong E'.

Using, for example, criterion (a) of Theorem 7, we see that, if E_1 and E_2 are locally convex spaces, and F_1 (resp. F_2) is a topological vector subspace of E_1 (resp. of E_2), then every integral bilinear form on F_1\times F_2 can be extended to a integral bilinear form on E_1\times E_2, with equal integral norm if E_1 and E_2 are normed (compare with Theorem 6). The most important properties of integral maps are summarised in the following:

Let u be an integral linear map from a locally convex space E to a locally convex space F.

  1. If F is quasi-complete, then u is weakly compact, and sends weakly compact subsets of E to compact subsets of F. If v is a linear map from F to a locally convex space G that sends bounded subsets to weakly relatively compact subsets, then v\circ u is a compact map.

  2. Let v be a linear map from F to an (\mathscr{F})-space G that sends bounded subsets to weakly relatively compact subsets (resp. a linear map from a (\mathscr{DF})-space G to E that sends bounded subsets to weakly relatively compact subsets of E). Then v\circ u (resp. u\circ v) is a nuclear map from E to G (resp. from G to F''). If E, F, and G are Banach spaces, then \|v\circ u\|'_1\leqslant\|v\|\|u\|'.

The composition of two integral maps is a nuclear map.

An integral map from E to F is nuclear if F is a reflexive (\mathscr{F})-space, and it is a nuclear map from E to F'' if E is a reflexive (\mathscr{DF})-space. An integral map from E to F sends summable sequences to absolutely summable sequences, and weakly convergent sequences to nuclearly convergent sequences (see the end of §1.4).

\star So, considering the unit circumference T of the complex plane, endowed with its Haar measure \mu, suppose that the sequence (a_n) in the set \mathbb{Z} of integers is such that (\varepsilon_n a_n) is the sequence of coefficients of a function in \mathrm{L}^\infty(\mu), for some sequence (\varepsilon) of numbers equal to +1 or -1; then we can easily see that the sequence a_nz^n\in\mathrm{L}^\infty(\mu) is summable, and is thus an absolutely summable sequence in \mathrm{L}^1(\mu), whence, immediately, (a_n)\in\ell^1. We have thus obtained an analogue of the well-known theorem of Littlewood (in which \mathrm{L}^\infty is replaced by \mathrm{L}^1, and \ell^1 by \ell^2). We note that Littlewood’s theorem can be obtained in the same way, as a consequence of the following theorem, which has much more general consequences (and which will later be published, along with some of its various consequences): every summable sequence in an \mathrm{L}^1(\mu)-space (for an arbitrary measure) has a sequence of norms that is square summable (and even belonging to \ell^2\hat{\otimes}\mathrm{L}^1(\mu); this is dual to the theorem, noted at the end of §1.4, that says that every square summable sequence in a \mathrm{C}(K)-space — and even every sequence belonging to \ell^2\hat{\hat{\otimes}}\mathrm{C}(K) — is nuclearly convergent to 0).

We will give some hints concerning the proof of Theorem 8, that rely in an essential manner on criterion (d) of Theorem 7. The fact that u is a weakly compact map follows immediately from the fact that the identity map from \mathrm{L}^\infty(\mu) to \mathrm{L}^1(\mu) is a weakly compact map. The other claims of the theorem follow easily from the second part, which is more difficult. We can reduce to showing that every weakly compact linear map from \mathrm{L}^1(\mu) to an (\mathscr{F})-space G induces a nuclear map from \mathrm{L}^\infty(\mu) to G, and thus (Theorem 3) comes from some f\in\mathrm{L}^1_G(\mu). But a theorem of Dunford–Pettis–Phillips tells us that such a map is in fact given by a strongly measurable and bounded map f from M to G. Note that Corollary 1, which is important due to its application in the theory of nuclear spaces, admits a simpler direct proof: we can reduce to showing that, if \mu and \nu are measures on compact M and N, then a continuous linear map u from \mathrm{L}^1(\mu) to \mathrm{L}^\infty(\nu) defines a nuclear map from \mathrm{L}^\infty(\mu) to \mathrm{L}^1(\nu). But u can be identified with a continuous linear form on \mathrm{L}^1(\mu)\hat{\otimes}\mathrm{L}^1(\nu), which is isomorphic to the space \mathrm{L}^1(\mu\otimes\nu) (Theorem 3), and so u is defined by a measurable and bounded function f on M\times N, which can be identified a fortiori with an element of \mathrm{L}^1(\mu\otimes\nu)=\mathrm{L}^1(\mu)\hat{\otimes}\mathrm{L}^1(\nu). This latter element indeed defines a nuclear map from \mathrm{L}^\infty(\mu) to \mathrm{L}^1(\nu), which happens to be exactly the map induced by u. QED.

In PTT, we deduce Theorem 8 from more general results (see [PTT, chap. 1, §4, no. 2]). The majority of [PTT, chap. 1, §4] (the densest part of the whole work) is dedicated to the exposition of these results and their many consequences, which we cannot give in this summary. \star

1.9 Integral linear maps to an \mathrm{L}^1 space or a \mathrm{C}_0(M) space

[PTT, chap. 1, §4, no. 4]

\star We can characterise the integral linear from a locally convex space E to an \mathrm{L}^1(\mu) space (where \mu is an arbitrary measure on a locally compact space M): they are the linear maps that send a suitable neighbourhood V of 0 in E to a latticially bounded subset of \mathrm{L}^1(\mu). If E is normed, with V its unit ball, and h=\sup_{x\in V}|ux| (so that h is a positive element of \mathrm{L}^1(\mu)), then \|u\|'_1=\|h\|_1. If E or \mathrm{L}^1(\mu) is separable, then the theorem of Dunford–Pettis gives an equivalent criterion (which we state, as an example, in the case where E is assumed to be normed): there exists a weakly measurable map f from M to E' such that \|f(t)\| is a summable function of t, and such that, for all x\in E, ux is the class in \mathrm{L}^1(\mu) of the function t\mapsto\langle x,f(t)\rangle; then \|u\|'_1\leqslant\int\|f(t)\|\mathrm{d}\mu(t) (and we have equality for a suitable choice of f). We can also characterise the nuclear maps from a locally convex space E to \mathrm{L}^1(\mu): they are those which send a suitable neighbourhood V of 0 in E to a subset A of \mathrm{L}^1(\mu) which is latticially bounded and further equimeasurable (by which we mean that, for every compact K\subset M, and every \varepsilon>0, there exists a compact K_0 such that \mu(K\cap K_0^c)\leqslant\varepsilon, and such that the \varphi\in A agree almost everywhere on K_0 with the functions from an equicontinuous and uniformly bounded set of functions on K_0). If we suppose, for simplicity, that E is a Banach space, then this is equivalent to saying that the map in question is given by an integrable map f from M to E' (and, indeed, it follows immediately from Theorem 3 that this indeed implies that u is nuclear).

Dually, let M be a locally compact space, and E a Banach space (for simplicity), and suppose that either M is metrisable and countable at infinity, or that E is separable. Then the integral maps u from \mathrm{C}_0(M) (the space of continuous functions on M that are “zero at infinity”) to E', i.e. the continuous linear forms on \mathrm{C}_0(M,E)=\mathrm{C}_0(M)\hat{\hat{\otimes}}E, are exactly those given by a measure \mu on M and a weakly \mu-measurable and bounded map f from M to E' by the formula u\varphi=\int\varphi(t)f(t)\mathrm{d}\mu(t). We can further suppose that \|f(t)\|=1 for all t, and that \|\mu\|_1=\|u\|'_1. Using Theorem 3, we also find that the nuclear maps from \mathrm{C}_0(M) to E', or even from \mathrm{C}_0(M) to an arbitrary Banach space F, are exactly those given by a pair (u,f) as above, but with f being an integrable map from M to F; it is no longer necessary here to make any separability hypotheses. In particular, if K and L are compact spaces, \mu a measure on K, and N(x,y) a continuous scalar function on K\times L, then the map f\mapsto\int f(x)N(x,y)\mathrm{d}\mu(x) from C(K) to C(L), defined by the continuous kernel N, is nuclear. Note that we have just described two remarkable categories of “vectorial measures” on M, which we call integral vectorial measures and nuclear vectorial measures (respectively) on M.

The fact that we can characterise the integral and nuclear maps from a Banach space (for example) E to \mathrm{L}^1(\mu) by properties concerning their images of the unit ball is completely special to \mathrm{L}^1(\mu) (see the end of §1.7) and is also linked to the corollary of Theorem 3. Similarly, we can show that the integral linear maps from \mathrm{L}^1(\mu) to a locally convex space E can be characterised by properties concerning their images of the unit ball of \mathrm{L}^1(\mu), a feature that we will not further study here (see [PTT, chap. 1, §4, no. 6]). \star

Appendix 1: Various variants of the notion of topological tensor product

[PTT, chap. 1, §3]

On E\otimes F, we can introduce a large number of distinct interesting topologies (even when E and F are Banach spaces). We will content ourselves here with noting the existence of a unique locally convex topology on E\otimes F such that, for every locally convex space G, separately continuous bilinear maps from E\times F to G correspond exactly to continuous linear maps from E\otimes F to G. To separately equicontinuous sets of bilinear maps from E\times F then correspond equicontinuous sets of linear maps from E\otimes F. Endowed with this topology, E\otimes F is called the inductive tensor product of E and F, and its completion, denoted by E\bar{\otimes}F. Its dual is thus the space \mathscr{B}(E,F), and the equicontinuous subsets of this dual are the separately equicontinuous sets of bilinear forms on E\times F (which already suffice to determine the topology of the inductive tensor product). The inductive tensor product topology on E\otimes F is finer than the projective tensor product topology, and these two topologies are identical if and only if the separately equicontinuous sets of bilinear forms on E\times F are already equicontinuous (for example, if E and F are of type (\mathscr{F}), or if E and F are of type (\mathscr{DF}) and barrelled). So if E is a non-normable space, then the two topologies above on E\otimes E' give distinct duals (since the canonical bilinear form on E\times E' is separately continuous but not continuous). — If E is the inductive limit (in the general sense) of a family of spaces E_i, and F the inductive limit of a family of spaces F_j, then the topological-vectorial subspace H of E\bar{\otimes}F generated by the canonical images of the spaces E_i\bar{\otimes}F_j is the inductive limit of these latter spaces (whence the name “inductive” tensor product). Unfortunately, the space H above is often not complete, i.e. it is distinct from E\bar{\otimes}F. — The analogous statement of the above, for when E and F are product spaces, is only true under fairly restrictive conditions, e.g. if F is the topological-vectorial product of a family of spaces of type (\mathscr{F}). — Finally, note that the notion of topological tensor product that we have just developed gives rise to a notion of tensor product of two continuous linear maps, completely analogous to the notion developed in §1.6.

We should define a remarkable dense subset of E\bar{\otimes}F, which is more important that E\bar{\otimes}F itself (even though it is often not complete, since it is distinct from E\bar{\otimes}F): it is the subspace given by the union of the canonical images of the spaces E_A\bar{\otimes}F_B=E_A\hat{\otimes}F_B, where A (resp. B) is a bounded disk of E (resp. F) such that E_A (resp. F_B) is complete. The elements of this subspace are called Fredholm kernels in E\bar{\otimes}F, and they appear, for example, in the theory of nuclear spaces (see Corollary 3 of Theorem 1 in §2.2). The subspaces of Fredholm kernels are sent to one another by tensor products of continuous linear maps. — Theorem 2 (in §1.2) shows that, if E and F are of type (\mathscr{F}), then every element of E\bar{\otimes}F=E\hat{\otimes}F is already a Fredholm kernel; it further gives an explicit structure theorem for Fredholm kernels in the general case: Fredholm kernels in E\bar{\otimes}F can be represented by series u = \sum\lambda_i x_i\otimes y_i where (x_i) (resp. (y_i)) is a series extracted from a compact disk of E (resp. F), and where (\lambda_i) is a summable sequence of scalars. Thus u itself comes from an element of some space E_A\hat{\otimes}F_B, where A and B are compact disks.

We define a Fredholm map from E to F to be a map defined by a Fredholm kernel of E'\bar{\otimes}F. Such a map is weakly continuous, but not necessarily continuous; but if every strongly compact disk of E' is equicontinuous (in particular, if the topology on E is the Mackey topology \tau(E,E')), then every Fredholm map from E to F is already nuclear, and a fortiori continuous. — Note that it is the Fredholm kernels of E'\bar{\otimes}E that form the natural domain for Fredholm theory.

Let E be a locally convex space, and take a locally convex topology on its dual that is less fine than the weak topology. Then the canonical bilinear form on E\times E' is separately continuous, and thus defines a continuous linear form on E\bar{\otimes}E', called the trace form. Let F be another locally convex space; let A\in\mathscr{B}(E,F), and suppose that the corresponding linear map {}^tA from F to E' is continuous (which will be the case, for example, if we take the strong topology on E' and assume that F is barrelled); then 1\otimes{}^tA is a continuous linear map from E\bar{\otimes}F to E\bar{\otimes}E', also denoted by u\mapsto {}^tAu. Then the passage to the trivial limit gives, for all u\in E\bar{\otimes}F, \langle u,A\rangle = \operatorname{Tr}{}^tAu.

This makes the duality between E\bar{\otimes}F and \mathscr{B}(E,F) explicit by means of the trace form. This formula immediately generalises for the natural pairing corresponding to any “reasonable” type of completed topological tensor product (PTT, §3, no. 3, prop. 17). — If K is a compact space endowed with a measure \mu, then we have already seen (cf. §1.9) that the integral operator defined by a continuous kernel N(x,y) (defined on K\times K) is a nuclear operator in \mathrm{C}(K). We can easily show that its trace is exactly \int N(x,x)\mathrm{d}\mu(x).

Appendix 2: The properties and problems of approximation

[PTT, chap. 1, §5]

The most important problem that remains to be solved in the theory of topological tensor products is the following “bijectivity problem”: is the canonical map from E\hat{\otimes}F to E\hat{\hat{\otimes}}F always bijective? By the Hahn–Banach theorem, this problem can be turned into of the variations of the “approximation problem”: is every continuous bilinear form on E\times F the limit, in the weak topology defined by E\hat{\otimes}F, of degenerate continuous bilinear forms, i.e. of those coming from E'\otimes F'? Under this form of the problem, we see that we can reduce to the case where E and F are Banach spaces. Since then the bicompact-convergence topology (i.e. uniform convergence on products of a compact of E with a compact of F) on \mathrm{B}(E,F) gives the dual E\hat{\otimes}F (cf. §1.2, Theorem 2), we can replace the weak topology defined by E\hat{\otimes}F in the “approximation problem” with the bicompact-convergence topology. This also proves that we can assume E and F to be separable. By continuing with such procedures (notably the systematic use of Theorem 2), we have given, in [PTT, chap. 1, §5, prop. 37], a large number of other equivalent formulations of the above conjecture. It suffices, for example, to assume in the above that E is a topological-vectorial subspace of c_0, and that F is its dual. It even suffices to prove that, if u\in E'\hat{\otimes}E defines a zero nuclear operator, then \operatorname{Tr}u=0. A more concrete formulation of this latter statement is the following: let u=(u_{ij}) be a matrix that represents an element of \ell^1\hat{\otimes}c_0 (i.e. such that \sum_i\sup_j|u_{ij}|<+\infty), and such that u^2=0; then \operatorname{Tr}u=\sum u_{ii}=0. Another formulation: let K(x,y) be a continuous kernel on X\times X (where X is a compact space endowed with a positive measure \mu) such that K\circ K=0; then \operatorname{Tr}K=\int K(x,x)\mathrm{d}\mu(x)=0. Or: let f(x,y) be a continuous function on the product of two compact spaces X and Y; then f is the uniform limit of linear combinations of functions of the type f(x,b)f(a,y). In these latter two examples, it suffices to do the proof in only one case for which the general conjecture has been proven, provided that \mu is not the sum of a sequence of discrete masses in the first case, or that X and Y are infinite in the second case. Other formulations are given in what follows.

We say that a locally convex space E satisfies the approximation condition if the identity map from E to itself is the limit, in the topology given by uniform convergence on every precompact subset, of continuous linear maps of finite rank. Then, for every locally convex space F, every continuous linear map from E to F, or from F to E, is the limit, in the precompact-convergence topology, of continuous linear maps of finite rank. If E is a Banach space, then this also implies that, for every Banach space F, every compact linear map from F to E is the limit, in the sense of the norm, of continuous linear maps of finite rank; or even that, for every Banach space G, the space G\hat{\hat{\otimes}}E is identical to the space of compact and weakly continuous linear maps from G' to E. We can show, by using Theorem 2, that this is equivalent to the fact that, for every Banach space F, the canonical map from E\hat{\otimes}F to \mathrm{B}(E',F') is bijective; and, in this condition, it suffices to take F=E' (and we thus have here a bijectivity condition), and even to suppose that the trace of any u\in E'\hat{\otimes}E that defines a zero operator is zero. — We can show that the dual E' of a Banach space E satisfies the approximation condition if and only if every compact linear map from E to a Banach space F is the limit, in the sense of the norm, of continuous linear maps of finite rank, and that this implies that E itself satisfies the approximation condition. — The “approximation problem” described above can also be stated as follows: does every locally convex space satisfy the approximation condition? We have already seen that we can restrict to closed vector subspaces of c_0.

The spaces \mathrm{L}^p (for 1\leqslant p\leqslant+\infty), constructed from an arbitrary measure, the spaces \mathrm{C}(K) (of continuous functions on a compact space K), as well as the duals, biduals, etc. of these spaces, all satisfy the approximation condition (and even the stronger “metric approximation condition”; see below). Nuclear spaces satisfy the approximation condition (see §2), and so too, more generally, do spaces that are isomorphic to subspaces of products of Hilbert spaces (these types of spaces arising rather frequently in practice). I give other examples in [PTT, chap. 1, §5, no. 3], including, notably, the most important amongst the Banach spaces given by distributions over \mathbb{R}^n (essentially those that are between (\mathscr{D}) and (\mathscr{D}') and that are stable under translation and under multiplication by \varphi\in(\mathscr{D})).

We say that a Banach space E satisfies the metric approximation condition if the identity map from E to itself is the limit, uniform on every compact subset, of linear applications of finite rank and of norm \leqslant 1. This is a metric strengthening of the approximation condition, and we can give analogous reformulations ([PTT, chap. 1, §5, no. 2]). We note the following: for every Banach space F, the canonical map from E\hat{\otimes}F to the space \mathrm{J}(E',F') of integral bilinear forms on E'\times F' is a metric isomorphism. It again suffices to prove that the canonical map from E\hat{\otimes}E' to \mathrm{J}(E',E) is a metric isomorphism. We do not know of any Banach space that does not satisfy the metric approximation condition.

We can again prove that E' satisfies the metric approximation condition if and only if, for every Banach space F, the canonical map from E'\hat{\otimes}F' to the space \mathrm{J}(E,F) of integral bilinear forms on E\times F is a metric isomorphism; we can then prove that E then also satisfies the metric approximation condition. More profound is the following result:

Let E, F, and G be Banach spaces, and u (resp. v) a continuous linear map from E to F (resp. from F to G). Suppose that one of the maps (u or v) is weakly compact, and that the other is the uniform limit, on every compact subset, of continuous linear maps of finite rank. Then w=v\circ u is the uniform limit, on every compact subset, of continuous linear maps of finite rank and of norm \leqslant\|w\|.

Let E be a reflexive Banach space. For E to satisfy the metric approximation condition, it suffices for it to satisfy the approximation condition.

These statements give conclusions of a metric nature from purely topological hypotheses, and thus hold true if we replace the given norms by equivalent norms. In this respect, the corollary to Theorem 9 gives, even for a Hilbert space, a new approximation result.

2 Nuclear spaces

2.1 Introduction to nuclear spaces

In the majority of examples where E is a given concrete complex locally convex space (a space of functions, or distributions, for example), and F is an arbitrary complete locally convex space, we do not know how to concretely, simultaneously characterise E\hat{\otimes}F, E\hat{\hat{\otimes}}F, and its topological-vectorial extension \mathscr{B}_e(E'_s,F'_s) (for example, interpret these spaces as spaces of functions or distributions with vector values, characterised in a simple way). But we do often know how to concretely describe a locally convex space P sitting between E\hat{\otimes}F and E\hat{\hat{\otimes}}F, or at least between E\hat{\otimes}F and \mathscr{B}_e(E'_s,F'_s) (“sitting between” meaning that the maps E\hat{\otimes}F\to P and P\to\mathscr{B}_e(E'_s,F'_s) are continuous).

Let E=\mathrm{L}^p(\mu), where 1\leqslant p<+\infty; if F is a Banach space (and, by extension, if F is an aribtrary complex locally convex space), the we define the space \mathrm{L}_F^p(\mu) of “p-th power integrable” maps from the measured space M to F (cf. [2]) in a natural way, and we easily see that \mathrm{L}^p\hat{\otimes}F \subset \mathrm{L}_F^p \subset \mathrm{L}^p\hat{\hat{\otimes}}F. In general, \mathrm{L}_F^p is different from \mathrm{L}^p\hat{\otimes}F and \mathrm{L}^p\hat{\hat{\otimes}}F, and the elements of \mathrm{L}^p\hat{\otimes}F do not admit an obvious internal characterisation as maps from M to F (unless p=1, in which case see §1, Theorem 3), whereas the elements of \mathrm{L}^p\hat{\hat{\otimes}}F can not even be interpreted, in general, as measurable or scalar-measurable maps from M to F.

For spaces E consisting of actual functions (and not only of classes of functions, like \mathrm{L}^p), we can often characterise \mathscr{B}_e(E'_s,F'_s), thanks to the following:

Let E be a function space on a set M, endowed with a locally convex topology that is finer than the simple-convergence topology. Then, for every complete locally convex space F, we can interpret \mathscr{B}_e(E'_s,F'_s) as the space of maps f from M to F such that, for all y'\in F', the function f_{y'}(t)=\langle f(t),y'\rangle belongs to E (i.e. f belonging scalar-wise to E), and such that f_{y'} runs over a weakly relatively compact subset of E as y' runs over an equicontinuous subset of F'. (This second condition is excessive if E is reflexive and of type (\mathscr{F}) or type (\mathscr{LF})).

But in a case such as the above, we do not, in general, have a handle on the space E\hat{\otimes}F.

We thus appreciate the simplifications that arise in the case where we know in advance that E\hat{\otimes}F=E\hat{\hat{\otimes}}F, or even that E\hat{\otimes}F=\mathscr{B}_e(E'_s,F'_s) (with these equalities being assumed to involve the topologies). Then we can often concretely determine E\hat{\otimes}F=E\hat{\hat{\otimes}}F=\mathscr{B}_e(E'_s,F'_s); for example, if we know a priori an intermediate space P between E\hat{\otimes}F and \mathscr{B}_e(E'_s,F'_s). At the same time, we will then have determined the space of weakly continuous linear maps from E' to F, or from F' to E, since these spaces can in fact be identified with \mathscr{B}_e(E'_s,F'_s). And the dual \mathrm{B}(E,F) of E\hat{\otimes}F, i.e. the space of continuous bilinear forms on E\times F, will then be identical to the dual of E\hat{\hat{\otimes}}F, or also to the dual of P, and can thus often be concretely understood.

For example, the essence of L. Schwartz’s “kernel theorem” says that the space of continuous bilinear forms on \mathscr{E}(\mathbb{R}^m)\times\mathscr{E}(\mathbb{R}^m) is identical to the space of bilinear forms defined by the distributions on \mathbb{R}^m\times\mathbb{R}^n with compact support. The latter are a priori the continuous linear forms on \mathscr{E}(\mathbb{R}^m\times\mathbb{R}^n), but we directly see (as a particular case of the above lemma) that \mathscr{E}(\mathbb{R}^m\times\mathbb{R}^n) can be identified with the space E\hat{\hat{\otimes}}F = \mathscr{B}_e(E'_s,F'_s) where E=\mathscr{E}(\mathbb{R}^m) and F=\mathscr{E}(\mathbb{R}^n), and so a priori the distribution on \mathbb{R}^m\times\mathbb{R}^n are the integral (§1.8) bilinear forms on \mathscr{E}(\mathbb{R}^m)\times\mathscr{E}(\mathbb{R}^m). The kernel theorem, which says that all continuous bilinear forms on \mathscr{E}(\mathbb{R}^m)\times\mathscr{E}(\mathbb{R}^m) are obtained in this way, i.e. that the transpose of the canonical map from \mathscr{E}(\mathbb{R}^m)\hat{\otimes}\mathscr{E}(\mathbb{R}^m) to \mathscr{E}(\mathbb{R}^m)\hat{\hat{\otimes}}\mathscr{E}(\mathbb{R}^m) is a surjective map, also implies (by a classical theorem of (\mathscr{F})-spaces) that we in fact have that \mathscr{E}(\mathbb{R}^m)\hat{\otimes}\mathscr{E}(\mathbb{R}^m) = \mathscr{E}(\mathbb{R}^m)\hat{\hat{\otimes}}\mathscr{E}(\mathbb{R}^m). In fact, the proof of the theorem even shows that \mathscr{E}(\mathbb{R}^m)\hat{\otimes}F = \mathscr{E}(\mathbb{R}^m)\hat{\hat{\otimes}}F (a space which is isomorphic, when F is complete, to the space of infinitely differentiable maps from \mathbb{R}^m to F) for every locally convex space F.

We thus see that, in general, the claim that E\hat{\otimes}F=E\hat{\hat{\otimes}}F for two given spaces E and F should be seen as an algebraic-topological equivalent of L. Schwartz’s kernel theorem. In what follows, we will study the spaces E, such as \mathscr{E}(\mathbb{R}^m), for which we have E\hat{\otimes}F=E\hat{\hat{\otimes}}F for every locally convex space F: these are the nuclear spaces.

2.2 Characterisation of nuclear spaces

[PTT, chap. 2, §2, no. 1]

By §2.1, a locally convex space is said to be nuclear if, for every locally convex space F, the canonical map from E\hat{\otimes}F to E\hat{\hat{\otimes}}F is a topological-vectorial isomorphism from the first space to the second (or, equivalently, if these two spaces induce the same topology on E\otimes F). It actually suffices to check this condition only in the case where F is a Banach space. Further, E is nuclear if and only if its completion is nuclear.

Let E be a locally convex space. Then the following conditions are equivalent:

  1. E is nuclear.

  2. Every continuous linear map from E to a Banach space F is nuclear ((§1.7)[#section-1.7]).

  3. For every equicontinuous weakly closed disk A in E', there exists another B\supset A such that the identity map from E'_A to E'_B is nuclear.

If E is then complete, we have, for every complete locally convex space F, that E\hat{\otimes}F = E\hat{\hat{\otimes}}F = \mathscr{B}_e(E'_s,F'_s) (i.e. topological-vectorial isomorphisms).

\star The hard part of the proof consists of showing that (a) implies (c). By taking the dual of condition (a), we find that, for every equicontinuous weakly closed disk A of E', there exists another B\supset A such that the identity map from E'_A to E'_B is integral ((§1.8)[#section-1.8]). Applying the same result to B, and using the corollary of Theorem 8, we find (c). We note that we have only used the fact that, for every Banach space F, the canonical map from E\hat{\otimes}F to E\hat{\hat{\otimes}}F is a weak isomorphism. \star

Since a nuclear map is a fortiori compact, (c) implies the following:

Let E be a nuclear space. Then the bounded subsets of E are precompact (and E is even a Schwartz space, cf. [6]). A fortiori, if E is quasi-complete, then E is of type (\mathscr{M}), and thus reflexive.

We thus also conclude that a nuclear Banach space is of finite dimension. Furthermore:

Let E be a nuclear space, and F an arbitrary locally convex space. If F is quasi-complete, then every bounded linear map from E to F is nuclear. If E is barrelled (e.g. if E is of type (\mathscr{F})), then every bounded linear map from F to E' is nuclear.

Let E and F be locally convex spaces, with E nuclear. Then the continuous bilinear forms on E\times F are exactly the “nuclear” bilinear forms, i.e. those which come from an element of a space of the form E'_A\hat{\otimes}F'_B, where A (resp. B) is an equicontinuous weakly closed disk in E' (resp. F').

We do not know of any case where we have E\hat{\otimes}F=E\hat{\hat{\otimes}}F without having either E or F already nuclear (in this problem, we can easily reduce to the case where E and F are both of type (\mathscr{F})). We can show that, if F is the space c_0 or \ell^1, or, more generally, if F admits a quotient space isomorphic to either c_0 or \ell^1, then the statement that E\hat{\otimes}F=E\hat{\hat{\otimes}}F already implies that E is nuclear. Combining this with Banach’s classical “isomorphism theorem,” we obtain:

Let E be an (\mathscr{F})-space. For E to be nuclear, it is necessary and sufficient for every summable sequence in E to be absolutely summable, or also for every sequence in E that converges to 0 to converge to 0 nuclearly (cf. the end of (§2.4)[#section-2.4]).

In particular, if E is a Banach space, then each of the two above conditions imply that E is of finite dimension: in the first case, we thus recover a recent theorem of Dvoretzky–Rogers.

2.3 Invariance properties, examples of nuclear spaces

[PTT, chap. 2, §2, no. 2 and no. 3]

The class of nuclear spaces enjoys a remarkable stability property, expressed in the following:

  1. Let E be an (\mathscr{F})-space or a (\mathscr{DF})-space. Then, for E to be nuclear, it is necessary and sufficient for E' to be nuclear.
  2. Let E be a nuclear space. Then every vector subspace and every quotient space of E is nuclear.
  3. The topological-vectorial product of an arbitrary family of nuclear spaces, and the topological-vectorial sum of a countable family of nuclear spaces, is nuclear.
  4. Let E and F be two nuclear spaces. Then E\hat{\otimes}F is nuclear.
  5. Let E be a nuclear space of type (\mathscr{F}), and F a reflexive space whose dual is nuclear. Then the dual of E\hat{\otimes}F is nuclear.

\star Of these claims, (2) and (5) are the most difficult (the others follow reasonably simply from the criteria of Theorem 1). The proof given in [PTT], which relies on rather acute techniques, can be replaced by the systematic use of criterion (c) of Theorem 1, along with the two following facts:

  1. If E is a nuclear space, then there exists a fundamental family of equicontinuous subsets A of E' such that E'_A is a Hilbert space (i.e. E is isomorphic to a vector subspace of the product of a family of Hilbert spaces);
  2. The remark made at the end of §1.7. \star

From Theorem 3, we obtain the following results: if E is a vector space endowed with the finest topology making some family of linear maps f_i from E to nuclear spaces E_i continuous, then E is nuclear; if E is a vector space, and u_i are linear maps from a sequence of nuclear spaces E_i to E such that E is generated by the union of the u_i(E_i), and if endow E with the finest locally convex topology making the u_i continuous, then E is nuclear; if E is a nuclear space of type (\mathscr{F}) or (\mathscr{DF}), and F is a nuclear space, then \mathrm{L}_b(E,F) is nuclear, and so too is the dual of \mathrm{L}_b(E,F) if F is also of type (\mathscr{F}) or (\mathscr{DF}).

Theorem 3, along with the fact that the space \mathscr{E}(U) of infinitely differentiable functions on an open subsets U of \mathbb{R}^n is nuclear (§2.1), allows us to easily prove the following:

All of the following are nuclear: the spaces \mathscr{E}, \mathscr{E}', \mathscr{D}, and \mathscr{D}', constructed on an open subset U of \mathbb{R}^n; the spaces (\mathscr{S}), (\mathscr{S}'), (\mathscr{O}_M), and (\mathscr{O}'_C), constructed on \mathbb{R}^n; and the strong duals of these latter two spaces. The space \mathscr{H} of holomorphic functions on a complex-analytic manifold is also nuclear. (For the definitions of the above spaces, see [9]).

\star Consider an increasing sequence of positive sequences a^{(n)}=(a_i^{(n)}), and let E be the corresponding “echelon” space (see [7]), i.e. the space of sequences (x_i) such that \sum_i x_ia_i^{(n)} is absolutely summable for all n. Note that E is endowed with a natural topology of an (\mathscr{F}) space. For E to be nuclear, it is necessary and sufficient for there to exist, for all n, some m\geqslant n such that the sequence a^{(n)}/a^{(m)}=(a_i^{(n)}/a_i^{(m)}) to be summable (where we take a quotient of two zero terms to be zero). Thus the space (s) of rapidly decreasing sequences, as well as its dual (s') (the space of slowly increasing sequences), is nuclear. (This easily recovers, thanks to the Fourier transform, the fact that \mathscr{E}(\mathbb{R}^n), and, more generally, \mathscr{E}(U), is nuclear.) The above statement also allows us to construct echelon spaces that are of type (\mathscr{M}) (and even of type (\mathscr{S}), see [6, Section 3]) but not nuclear. \star

2.4 Lifting properties

[PTT, chap. 2, §3, no. 1]

Theorem 6 of §1.7 can be specialised as follows:

  1. Let E_1 and E_2 be locally convex spaces, and F_1 (resp. F_2) a vector subspace of E_1 (resp. E_2). Suppose that F_1 or F_2 is nuclear. Then every continuous bilinear form on F_1\times F_2 is the restriction of a nuclear bilinear form on E_1\times E_2.

  2. Let E and G be locally convex spaces, and F a vector subspace of E. Suppose that F is nuclear and that G is quasi-complete, or that G is the strong dual of a quasi-barrelled nuclear space (for example, that G is a complete nuclear space of type (\mathscr{F}) or (\mathscr{DF})). Then every bounded linear map from F to G is the restriction of a nuclear map from E to G.

  3. Let E and G be locally convex spaces, and F a closed vector subspace of E, such that every compact disk of E/F is contained in the canonical image of a bounded complete disk of E. Suppose that G is nuclear, or that E/F is isomorphic to the strong dual of a quasi-barrelled nuclear space (for example, that E is a complete nuclear space of type (\mathscr{F}) or (\mathscr{DF})). Then every bounded linear map from G to E/F comes from a nuclear map from G to E.

A variant of the above properties is the following: Let E be a locally convex space, F a nuclear vector subspace, and A an equicontinuous disk in E'; then there exists a linear map from F'_A to E' that is right inverse to the canonical map from E' to F', and that sends A to an equicontinuous subset of E'. We note again the following “lifting” property, which is an easy consequence of the above results:

Let E and F be locally convex spaces, both of type (\mathscr{F}), or both of type (\mathscr{DF}). Suppose that E is nuclear, and let A be a bounded disk in P=E\hat{\otimes}F. Then there exists a sequence (x_i) that tends to 0 in E, an equicontinuous sequence of linear maps u\mapsto y_i(u) from P_A to F, and a fixed summable sequence (\lambda_i), such that u = \sum \lambda_i x_i\otimes y_i(u) for all u\in P_A.

The bounded disk A is contained in the closed disked hull of a set B\otimes C, where B (resp. C) is some bounded subset of E (resp. F). Thus, on \mathrm{B}(E,F), the bibounded convergence topology and the topology of the strong dual of E\hat{\otimes}F are identical.

We note again that the analogue of Theorem 5 for the representation of convergent sequences in E\hat{\otimes}F also holds true.

2.5 The tensor product of a nuclear space with an arbitrary locally convex space

[PTT, chap. 2, §3, no. 2 and no. 3]

The duality theory for E\hat{\otimes}F is particularly simple when E is nuclear, and when E and F are both of type (\mathscr{F}), or both of type (\mathscr{DF}). In fact, the above results easily give:

Let E and F be locally convex spaces, both of type (\mathscr{F}), or both of type (\mathscr{DF}). If E is nuclear, then the strong dual of E\hat{\otimes}F, which can be identified with \mathrm{B}(E,F) endowed with the bibounded convergence topology (cf. the corollary of Theorem 5), can also be identified with E'\hat{\otimes}F'.

Subsequently, if E is complete, and thus reflexive, then the bidual of E\hat{\otimes}F, endowed with the strong topology of the dual of (E\otimes F)', can be identified with E\hat{\otimes}F''_b, where F''_b denotes the strong dual of F'.

In general, if E is nuclear and complete, but E and F are otherwise arbitrary, and if we again take the “natural” topologies on the biduals (cf. Introduction), then we can prove that the bidual of E\hat{\otimes}F can be identified with a dense topological-vectorial subspace of E\hat{\otimes}F'' (see [PTT, chap. 1, §4, no. 2, prop. 25], where we give a more general statement). As for the dual of E\hat{\otimes}F, whose elements are characterised by Corollary 3 of Theorem 1, it will, in general, no longer be identical to E'\hat{\otimes}F' at all, but, under rather strong conditions (such as if F is reflexive), it will be a dense topological-vectorial subspace of E'\bar{\otimes}F (see the definition in §1, Appendix 1). But even in the case of the tensor product E'\hat{\otimes}E, where E is nuclear and of type (\mathscr{F}) or type (\mathscr{DF}), the strong dual of E\hat{\otimes}F will often often not be complete (and thus distinct from E'\bar{\otimes}F'): see the appendix.

The product E\hat{\otimes}F (with E nuclear) gives rise to many invariance theorems: for E\hat{\otimes}F to be reflexive (resp. of type (\mathscr{M}), resp. of type (\mathscr{S}), resp. quasi-normable), it suffices (and is necessary, if F is complete) for F to be so. But it can be the case that E and F are bornological and barrelled without E\hat{\otimes}F being so (see the appendix).

By the comments of §2.1, concretely determining E\hat{\otimes}F when E is nuclear usually poses no difficulty. We now give some examples (where F is assumed to be a complete locally convex space).

Let \mathscr{E}=\mathscr{E}(V) be the space of infinitely differentiable functions on some given infinitely differentiable manifold V. Then \mathscr{E}\hat{\otimes}F is the space of infinitely differentiable functions from V to F (endowed with its natural topology). The analogous statement holds when we replace \mathscr{E} by the subspace E of functions satisfying some given system of partial differential equations \mathrm{D}_if=0.

Let \mathscr{H}=\mathscr{H}(V) be the space of holomorphic functions on some given complex-analytic manifold V. Then \mathscr{H}\hat{\otimes}F is the space of holomorphic maps from V to F (endowed with the compact convergence topology).

In particular, we find that \begin{aligned} \mathscr{E}(U)\hat{\otimes}\mathscr{E}(V) &= \mathscr{E}(U\times V) \\\mathscr{H}(U)\hat{\otimes}\mathscr{H}(V) &= \mathscr{H}(U\times V) \end{aligned} where U and V are some given infinitely differentiable (resp. holomorphic) manifolds.

Let \mathscr{S}=\mathscr{S}(\mathbb{R}^n) be the space of “infinitely differentiable and rapidly decreasing” functions on \mathbb{R}^n (cf. [9, t. 2]). Then \mathscr{S}\hat{\otimes}F is the space of maps from \mathbb{R}^n to F that are “infinitely differentiable and rapidly decreasing,” i.e. infinitely differentiable, and such that, for every derivation multi-index \mathrm{D}^p, and for every polynomial P on \mathbb{R}^n, the function P\mathrm{D}^pf is bounded on \mathbb{R}^n. In particular, \mathscr{S}(\mathbb{R}^m)\hat{\otimes}\mathscr{S}(\mathbb{R}^n) = \mathscr{S}(\mathbb{R}^m\times\mathbb{R}^n).

We have analogous statements given by taking E=(\mathscr{O}_M) (the space of “infinitely differentiable and slowly increasing” functions) or E=(\mathscr{O}_C) (the space of “infinitely differentiable and very slowly increasing” functions, the strong dual of the space (\mathscr{O}'_C) of rapidly decreasing distributions [9]), etc. We also note that, by using the Lemma in §1, we find that a map f with values in F is infinitely differentiable (resp. holomorphic, resp. “infinitely differentiable and rapidly decreasing,” resp. “infinitely differentiable and slowly increasing,” etc.) if and only if it is so “scalar-wise,” i.e. if and only if, for all y'\in F', the scalar-valued function \langle f(t),y'\rangle has the property in question.

2.6 p-th power summable operators, application to nuclear spaces

[PTT, chap. 2, §1, no. 1 to no. 6, and §2, no. 4]

Let E and F be locally convex spaces. Among the elements of of E\bar{\otimes}F (see the definition in §1, Appendix 1), we have given the name of Fredholm kernels to those which come from an element of some space E_A\hat{\otimes}F_B, where A (resp. B) is a bounded disk in E (resp. F) such that E_A (resp. F_B) is complete; we can then even suppose A and B to be compact (§1, Appendix 1). More generally, if 0<p\leqslant 1, then we call any element of E\bar{\otimes}F of the form \sum\lambda_i x_i\otimes y_i, where (\lambda_i)\in\ell^p and (x_i) (resp. (y_i)) is a sequence inside a compact subset of E (resp. F), a p-th power summable Fredholm kernel in E\bar{\otimes}F. (If p=1, then we recover Fredholm kernels). The set of p-th power summable Fredholm kernels in E\bar{\otimes}F is denoted by E\overset{(p)}{\otimes}F.

We define a p-th power summable map from E to F to be any linear map from E to F that comes from a p-th power summable Fredholm kernel in E'\bar{\otimes}F (if p=1, then we recover Fredholm maps, introduced in §1, Appendix 1). The space of these maps, denoted by \mathrm{L}^{(p)}(E,F), can thus be identified with a quotient space of E'\overset{(p)}{\otimes}F, and even with E'\overset{(p)}{\otimes}F in all known cases (see the “problem of bijectivity” in §1, Appendix 2); this is always the case, for example, if p\leqslant 2/3. When E and F are Banach spaces, we can introduce a “distance to the origin” function on E\otimes F, by S_p(u) = \inf\sum|\lambda_i|^p where the infimum is taken over all representations u=\sum\lambda_i x_i\otimes y_i with \|x_i\|,\|y_i\|\leqslant 1. Then E\overset{(p)}{\otimes}F becomes a complete metrisable topological-vectorial space which is, in general, not locally convex. When E and F are arbitrary locally convex spaces, the semi-norms of E and F allow us to define a system of “distances to the origin” on E\overset{(p)}{\otimes}F, which again make a topological-vectorial space (which is, in general, not locally convex); this space is metrisable and complete if E and F are metrisable and complete. Consequently, if E is of type (\mathscr{DF}), and F of type (\mathscr{F}), then \mathrm{L}^{(p)}(E,F) is a complete metrisable topological-vectorial space (which is, in general, not locally convex) isomorphic to a quotient space of E'\overset{(p)}{\otimes}F.

We can also, for 0\leqslant p<1, introduce the space E\overset{[p]}{\otimes}F of Fredholm kernels of order \leqslant p, which is defined to be the intersection of the spaces E\overset{(q)}{\otimes} F for p<q\leqslant 1. Endowed with the topology that is the upper bound of the topologies induced by the spaces E\overset{(q)}{\otimes} F, this is a topological-vectorial space (which is not, in general, locally convex), which is metrisable and complete if E and F are metrisable and complete. We analogously define the topological-vectorial spaces \mathrm{L}^{[p]}(E,F) as the intersection of the spaces \mathrm{L}^{(q)}(E,F) for p<q\leqslant 1. The introduction of these topologies allows us to prove the following: let E and F both be (\mathscr{F})-spaces; then every element of E\overset{[p]}{\otimes}F (where 0\leqslant p<1) is of the form \sum\lambda_i x_i\otimes y_i, where (\lambda_i)\in\ell^{[p]}, and where (x_i) (resp. (y_i)) is a bounded sequence in E (resp. F); further, if (u_\alpha) is a sequence that tends to 0 in E\overset{(p)}{\otimes}F (resp. in E\overset{[p]}{\otimes}F), then u_\alpha=\sum\lambda_i^\alpha x_i\otimes y_i, where the bounded sequences (x_i) and (y_i) are fixed, and where \lambda^\alpha=(\lambda_i^\alpha) tends to 0 in \ell^p (resp. \ell^{[p]}). (We denote by \ell^{[p]} the space given by the intersection of the spaces \ell^q for p<q\leqslant 1, endowed with the topology that is the upper bound of the topologies induced by the \ell^q, which in fact gives a complete metrisable space). The most important application of these results is the following:

Let E be a (\mathscr{DF})-space, and F an (\mathscr{F})-space. Then every map of order 0 from E to F is of the form \sum\lambda_i x'_i\otimes y_i, where (\lambda_i) is a rapidly decreasing sequence of scalars, and where (x'_i) (resp. (y_i)) is a sequence that tends to 0 in E' (resp. F).

Let E be an arbitrary locally convex space. Then every Fredholm kernel element of E'\bar{\otimes}E admits a Fredholm determinant \det(1-zu), which is an entire function in the complex variable z, and whose zeros are the inverses of the eigenvalues z_i of the Fredholm operator defined by u (where we count the zeros and the eigenvalues with multiplicity). We have the following:

If u\in E'\overset{(p)}{\otimes}E, then the Fredholm determinant of u is an entire function of order \leqslant q, where 1/q=1/p-1/2, and the sequence of eigenvalues of u is p-th power summable. If p\leqslant 2/3 (and so q\leqslant 1), then \det(1-zu) is an entire function of genus 0, and thus identical to the infinite product \prod(1-zz_i) (where the z_i are the eigenvalues of u, repeated according to their order of multiplicity). If E is a Banach space, then we have \left(\sum|z_i|^q\right)^{\frac1q} \leqslant(S_p(u))^{\frac1p}.

The sequence of eigenvalues of a Fredholm operator is square summable.

If u is a \frac23-summable Fredholm operator, then \operatorname{Tr}u=\sum_i z_i (where the z_i are the eigenvalues of u).

The sequence of eigenvalues of a Fredholm operator of order 0 in E, arranged in order of decreasing modulus, is a rapidly decreasing sequence.

Furthermore, we can show that, under the conditions of Theorem 8, when E is a Banach space, the sequence of eigenvalues of u\in E'\overset{(p)}{\otimes}E, as an unordered q-th power summable sequence, depends continuously on u. This implies that, if u_\alpha\to u in E'\overset{(p)}{\otimes}E, then we can arrange the eigenvalues of u_\alpha into a sequence \lambda^{(\alpha)}=(\lambda_i^{(\alpha)}) such that \lambda^{(\alpha)}\to\lambda^{(0)} in \ell^q. We have an analogous result for E'\overset{[p]}{\otimes}E.

By iterating Fredholm kernels, we obtain kernels that have stronger and stronger “decreasing properties”:

Let u be a Fredholm kernel given by the composition of n Fredholm kernels u_i, with each u_i being p_i-th power summable (with 0<p_i\leqslant 1). Then u is p-th power summable, where \frac1p = \left(\sum\frac{1}{p_i}\right)-\frac{n+1}{2}. If u_i\in E'_i\overset{(p_i)}{\otimes}E_{i+1}, where the E_i are Banach spaces, then S_p(u)^{\frac1p} \leqslant\prod\left(S_{p_i}(u_i)\right)^{\frac{1}{p_i}}.

In any case, the composition of n Fredholm operators (for n\geqslant 3) is is (2/(n-1))-th power summable. For n=2 or n=3, the formula 1/p=(\sum(1/p_i))-(n+1)/2 does not give any information, but it seems that we should replace n+1 by n-1 in it.

For more details on the above classes of operators, I refer the reader to [PTT, chap. 2, §1]. We note only that we can show that, up to a small numerical uncertainty in the value of the exponents (uncertainty being the very nature of things), the fact that, for an operator to be p-th power summable, it suffices to study the properties of the image of a suitable neighbourhood of 0 in E (the unit ball, if E is a Banach space). The uncertainty disappears for operators of order 0, which are characterised exactly by the fact that they send a suitable neighbourhood of 0 in E to a subset of F that is “of order 0” [PTT, chap. 2, §1, no. 5]. When F is of type (\mathscr{F}), the subsets of F of order 0 are the subsets contained in the closed disked hull of a “rapidly decreasing” sequence in F.

The combination of Theorem 1 and Theorem 9 gives:

Let E be a nuclear space. Then, for every \varepsilon>0, and every weakly closed equicontinuous disk A in E', there exists a weakly closed equicontinuous disk B\supset A such that the identity map from E'_A to E'_B is \varepsilon-th power summable.

Let E be a nuclear space, and F a locally convex space. Then every quasi-complete bounded linear map from E to F, and every linear map from F to E' that sends a suitable neighbourhood of 0 to an equicontinuous subset, is of order 0. Every continuous bilinear form on E\times F comes from an element of E'\overset{[0]}{\otimes}F' (i.e. is a Fredholm kernel of order 0).

Let E be a quasi-complete nuclear space. Then every bounded operator to E is a Fredholm operator of order 0, whose Fredholm determinant is thus of order 0, and the sequence of eigenvalues, arranged in order of decreasing modulus, is a rapidly decreasing sequence.

Other corollaries can be obtained by taking into account the other results given in this section. Thus, if E and F are both of type (\mathscr{F}), with E nuclear, then every element of E\hat{\otimes}F is of the form \sum\lambda_i x_i\otimes y_i, where (x_i) (resp. (y_i)) is a bounded sequence in E (resp. F), and where (\lambda_i) is a rapidly decreasing sequence. If E and F are of type (\mathscr{DF}), with E still nuclear, then we only assume that \sum|\lambda_i|^\varepsilon<+\infty, with \varepsilon>0 being given in advance. Similarly, Theorem 5 can be made more precise in an analogous way. More generally, if E and F are both locally convex spaces, with E nuclear, then every continuous bilinear form u on E\times F is of the form \sum\lambda_i x'_i\otimes y'_i, where (\lambda_i)\in\ell^\varepsilon, and where (x'_i) (resp. (y'_i)) is an equicontinuous sequence in E' (resp. F'). If u varies in a weakly closed equicontinuous disk M of bilinear forms, then we can take (x'_i) and (\lambda_i) in the above to be fixed, and y'_i=\varphi_i(u), where the \varphi_i form an equicontinuous sequence of linear maps from the Banach space generated by M to some space F'_B, where B is a weakly closed equicontinuous disk in F'. When F is also nuclear (it even suffices for it to be a Schwartz space), we can also take (x'_i) and (y'_i) in the above to be fixed, and \lambda=(\lambda_i) to be \lambda=\varphi(u), where \varphi is a continuous linear map from the Banach space generated by M to the space \ell^\varepsilon; we can even assume that the restriction of u to M endowed with the simple-convergence topology to be continuous, which gives another useful representation of equicontinuous bilinear forms on E\times F that tend to 0 (in the sense of simple convergence). Finally, we note that, if M is a set of endomorphisms of a nuclear space E, with each one sending a fixed neighbourhood of 0 to a fixed bounded subset of E, then the map that sends each u\in M to the unordered sequence of its eigenvalues is a continuous map from M, endowed with the simple-convergence topology, to the space of unordered rapidly decreasing sequences (endowed with its natural metrisable topology): this claim can be made explicit in the same way as the analogous claim found after Theorem 8.

2.7 On the spaces E\hat{\otimes}F with E of type (\mathscr{DF}) and F of type (\mathscr{F})

[PTT, chap. 2, §4]

The theory of duality for spaces of the form E\hat{\otimes}F, which is very simple when E is nuclear and E and F are both of type (\mathscr{F}), or both of type (\mathscr{DF}) (see Theorem 6), becomes more complicated in the general case. We can show, when E is nuclear, that the strong dual \mathrm{B}(E,F) of E\hat{\otimes}F can be identified with a dense vector subspace of E'\bar{\otimes}F', but with a topology that is a priori less fine (and sometimes strictly less fine) than the topology induced by E'\bar{\otimes}F'. When F is reflexive (for example), these topologies coincide; and when, further, E and F are both of type (\mathscr{F}), or both of type (\mathscr{DF}), then \mathrm{B}(E,F) is a (\mathscr{LF}) space, in the general sense (the generalised inductive limit of a sequence of spaces of type (\mathscr{F}), cf. [PTT, Introduction, IV]). But often, even if E and F are both nuclear, with E of type (\mathscr{DF}) and F of type (\mathscr{F}), the dual \mathrm{B}(E,F) of E\hat{\otimes}F is distinct from E'\bar{\otimes}F', i.e. \mathrm{B}(E,F) is not complete (a fortiori E\hat{\otimes}F is not then bornological). For example, if E is the direct topological sum of a sequence of lines, then the disagreeable situation described above arises whenever F is a non-normable (\mathscr{F}) space that admits a continuous true norm. In [PTT, chap. 2, §4, no. 1, prop. 14], we study a more general case, implying, for example, that, for the space \mathscr{E}'\hat{\otimes}\mathscr{E}=\mathrm{L}(\mathscr{E},\mathscr{E}) (where \mathscr{E}=\mathscr{E}(\mathbb{R}^n), which is a Schwartz space), the same troubles arise. Other important analogous spaces, such as \mathscr{S}'\hat{\otimes}\mathscr{S}=\mathrm{L}(\mathscr{S},\mathscr{S}) (where \mathscr{S}=\mathscr{S}(\mathbb{R}^n) is the space of “infinitely differentiable rapidly decreasing” functions on \mathbb{R}^n) are, however, bornological; but we do not have simple criteria that allow us to recognise, in general, if E\hat{\otimes}F is bornological or not. Without supposing E or F to be nuclear, if E is of type (\mathscr{DF}) and F of type (\mathscr{F}), then we can, under hypotheses that are more than sufficient in practice, we can confirm the equivalence of the following conditions:

  1. E\hat{\otimes}F is bornological;
  2. E\hat{\otimes}F is barrelled;
  3. \mathrm{B}(E,F) is complete;
  4. \mathrm{B}(E,F) is quasi-complete (or just complete for sequences).

It turns out that the hypotheses necessary for this theorem are satisfied when F is the the “echelon” space constructed by the procedure of G. Köthe [7] from an increasing sequence of positive sequences (b^n)=(b_i^{(n)}). When, further, E is the dual of a nuclear echelon space, or, more generally, if E is the inductive limit (in the general sense) of a sequence of normed spaces a^{(n)}\ell^1 (where, for a given sequence a^{(n)}=(a_i^{(n)}), we write a^{(n)}\ell^1 to denote the set of sequences given by a product of a^{(n)} with some summable sequence), then we can give an explicit exact criterion for E\hat{\otimes}F to be bornological: for E\hat{\otimes}F to be bornological, it is necessary and sufficient that, for each integer n_0>0, there exist an integer n>0 such that, for every integer m>0 and every positive sequence \lambda=(\lambda_i)\in E', there exists some R>0 such that, for every pair (i,j) of indices, one of the two following inequalities holds: \lambda_i a_i^{(m)}b_j^{(m)} \leqslant Rb_j^{(n)} \quad\text{or}\quad Ra_i^{(n)}b^{(n)} \geqslant a_i^{(m)}b_j^{(n_0)}. This statement is perfectly manageable in all particular cases, despite its barbaric aspect. In [PTT, chap. 2, §4, no .4], I apply this criterion to an interesting class of echelon spaces, including many interesting spaces in analysis. We find, for example, that, if F is the space of holomorphic functions on an open subset U of the complex plane, then F'\hat{\otimes}F=\mathrm{L}(F,F) is bornological if U=\mathbb{C}, but not bornological if U is the unit circle!

The most important consequence of this criterion is that the space (s')\hat{\otimes}(s), where (s) is the space of rapidly decreasing sequences, is bornological. Since (s) is isomorphic, by the Hermite transform, to the space \mathscr{S}(\mathbb{R}^n) of “infinitely differentiable rapidly decreasing” functions on \mathbb{R}^n, the space \mathscr{S}'\otimes\mathscr{S} is also bornological. We can, for example, thus conclude that the spaces (\mathscr{O}_M) and (\mathscr{O}'_C) of L. Schwartz are bornological [PTT, chap. 2, §4, no .4]. So, combining this with the results of §2.3, we obtain the following:

The spaces (\mathscr{O}_M) and (\mathscr{O}'_C) are bornological, complete, and nuclear, and their duals are bornological complete nuclear (\mathscr{LF}) spaces.

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  1. Such a formulation of Fredholm theory seems to have appeared for the first time in A. Ruston, “Direct product of Banach spaces and linear functional equations,” Proc. of the London Math. Soc. 3 (1951), 1. My work on this subject was conceived independently of his (in the autumn of 1951), and is rather different.↩︎

  2. This terminology was suggested to me by R.E. Edwards. [Trans.] The seemingly more popular terminology these days is to say “absolutely convex” instead of “disked,” and to speak of the “absolutely convex hull” instead of the “disked hull.”↩︎

  3. With respect to this point, we done that we can prove, by a completely different method, the following claim, which can be thought of as dual to the corollary of Theorem 3: Let M be a locally compact and paracompact space (e.g. a compact space), and f a continuous map from M to a quotient space E/F of an (\mathscr{F})-space E. Then f comes from a continuous map from M to E. Since the space \mathscr{E}^{(m)}(V) of m-times continuously differentiable functions on a infinitely differentiable paracompact manifold V is isomorphic to a direct factor of a space of the form C(M) (as I noted in “Sur les applications linéaires faiblement compacts d’espaces du type C(K),” Can. J. Math. 5 (1953), p. 144), it thus follows that the analogous lifting theorem holds also for m-times continuously differentiable maps from V to E/F.↩︎