# Summary of essential results in the theory of topological tensor products and nuclear spaces

*1952*

#### 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: `f36a214`

# 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,

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

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}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the spaces*, and their topological-vectorial analogues (these results being, as of yet, not definitive, and thus unpublished).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].L^1 ,L^2 , andL^\infty 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.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 *closed* vector subspace.
The *dual* of *bidual* of * (\mathscr{DF})-space*.
For our purposes here, it will suffice to know that the dual of a

Let *biequicontinuous topology*, i.e. the topology given by uniform convergence on the products of an equicontinuous subset of

We define a *bounded* (resp. *compact*, resp. *weakly compact*) *linear map* from

For short^{2}, if *disk* or *disked set in E* to mean a convex and circled (a.k.a. balanced) subset of

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

# 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

If *continuous* bilinear maps from *continuous* linear maps from

Then the *equicontinuous* subsets of *equicontinuous* subsets of *projective tensor product of the topologies of E and F*;
endowed with this topology,

*projective topological tensor product*of

If *normed* space

We can introduce the completion of *completed projective tensor product* of *scholium*: if *complete* locally convex space, then the continuous bilinear maps from

This claim still holds true for equicontinuous sets of maps.
In particular, the dual of

I do not know if, when *topological* isomorphism, when we endow

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

[PTT, chap. 1, §2, no. 1]

Let

(It is also true that, if *characterisation* of the elements of

Let

This fact also implies that, on

*onto* the latter, whence it indeed follows that

From Theorem 2, we can extract results of the following type:
let

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

[PTT, chap. 1, §2, no. 2]

Let

The above map from

To see this, we can immediately reduce to the case where

If

If

This recovers, for example, the well-known fact that every continuous linear map from

## 1.4 Other examples

If *Fredholm maps* or *nuclear maps* from *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

Numerous other examples of products *nuclear spaces*, will be seen in §2.5.

*certain* sequences in *nuclearly convergent to 0*;
but if

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

[PTT, chap. 1, §3, no. 3]

If

The topology on

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 *onto* the second), then

Let

Let

As with the spaces *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

## 1.6 Tensor product of linear maps

[PTT, chap. 1, §1, no. 2]

Let

If each

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

If the *onto* *onto*

As particular cases of this corollary, we obtain interesting lifting properties of vector functions with values in a quotient space of an ^{3}
Another application is the following:
let *onto* topological homomorphism), then so too is

We note that, if *onto* topological homomorphisms).
If each *one single* bilinear form.
In general, this criterion will not hold true, but it is linked to an existence problem of topological complements.

A useful case where the tensor product *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*.

## 1.7 Nuclear maps

[PTT, chap. 1, §3, no. 2]

Let *nuclear maps* from *trace maps* and *Fredholm maps* — see Appendix 1 — which coincide with the notion of nuclear maps if *trace norm* of the nuclear operator

If *nuclear* if it is the composition of a sequence of three operators
*a priori*, *A nuclear map is always compact* (i.e. sends a suitable neighbourhood of *compact* subsets of *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

Nuclear maps from *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

Every nuclear map from

F toG is the restriction of a nuclear map fromE toG .Suppose that

F is closed, and that every compact disc ofE/F is contained in the canonical image of a bounded discA ofE such thatE_A is complete (for example, a*complete*bounded disc). (It suffices, for example, forE to be an(\mathscr{F}) -space, or for it be the dual of an(\mathscr{F}) -space and forF to be*weakly*closed.) Then every nuclear map fromG toE/F can be obtained from a nuclear map fromG toE 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

We note that, if *is not nuclear*, even if

## 1.8 Integral linear maps, integral bilinear forms

[PTT, chap. 1, §4, no. 3 and no. 4]

Let

u\mapsto\langle u,v\rangle is a linear form onE\otimes F that is continuous for the topology induced byE\hat{\hat{\otimes}}F (resp.u\mapsto\langle u,v\rangle is a linear form onE\otimes F of norm\leqslant 1 whenE\otimes F is endowed with the norm induced byE\hat{\hat{\otimes}}F ).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 setM\otimes N , whereM is an equicontinuous subset ofE' , andN an equicontinuous subset ofF' (resp.M the unit ball ofE' , andN the unit ball ofF' ).There exists a measure

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

\mu of norm\leqslant 1 , a continuous linear map\alpha fromE to\mathrm{L}^\infty(\mu) , and a continuous linear map\beta fromF to\mathrm{L}^\infty(\mu) (resp. the same, but with\alpha and\beta also of norm\leqslant 1 ) such that we haveu(x,y)=\langle\alpha x,\beta y\rangle forx\in E ,y\in F .

A bilinear form on *integral* if it satisfies any of the equivalent conditions of Theorem 7.
In particular, the dual of *integral norm*, and denote by *integral* if the corresponding bilinear form on

If

Recall that, in all known cases, when *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

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

Using, for example, criterion (a) of Theorem 7, we see that, if

Let

If

F is quasi-complete, thenu is weakly compact, and sends weakly compact subsets ofE to compact subsets ofF . Ifv is a linear map fromF to a locally convex spaceG that sends bounded subsets to weakly relatively compact subsets, thenv\circ u is a compact map.Let

v be a linear map fromF to an(\mathscr{F}) -spaceG that sends bounded subsets to weakly relatively compact subsets (resp. a linear map from a(\mathscr{DF}) -spaceG toE that sends bounded subsets to weakly relatively compact subsets ofE ). Thenv\circ u (resp.u\circ v ) is a*nuclear*map fromE toG (resp. fromG toF'' ). IfE ,F , andG 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

*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

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 *strongly measurable and bounded* map *nuclear* map from

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.

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

[PTT, chap. 1, §4, no. 4]

*latticially bounded* subset of *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

*equimeasurable*(by which we mean that, for every compact

*integrable*map

Dually, let *nuclear* maps from *integrable* map from *integral vectorial measures* and *nuclear vectorial measures* (respectively) on

The fact that we can characterise the integral and nuclear maps from a Banach space (for example)

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

[PTT, chap. 1, §3]

On *separately continuous* bilinear maps from *continuous* linear maps from *inductive tensor product* of *product* spaces, is only true under fairly restrictive conditions, e.g. if *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 *Fredholm kernels* in *compact* disks.

We define a *Fredholm map* from

Let *trace* form.
Let

This makes the duality between

## 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 *approximation problem*”:
is every continuous bilinear form on *one* case for which the general conjecture has been proven, provided that

We say that a locally convex space *approximation condition* if the identity map from *every* locally convex space *every* continuous linear map from *norm*, of continuous linear maps of finite rank;
or even that, for every Banach space *bijectivity condition*), and even to suppose that the trace of any

The spaces

We say that a Banach space *metric approximation condition* if the identity map from *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

We can again prove that

Let *and of norm \leqslant\|w\|*.

Let

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

Let

For spaces

Let *scalar-wise* to

But in a case such as the above, we do not, in general, have a handle on the space

We thus appreciate the simplifications that arise in the case where we know in advance that

For example, the essence of L. Schwartz’s “kernel theorem” says that the space of continuous bilinear forms on *integral* (§1.8) bilinear forms on *all* continuous bilinear forms on *surjective* map, also implies (by a classical theorem of *every* locally convex space

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

*every*locally convex space

*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

Let

E is nuclear.Every continuous linear map from

E to a Banach spaceF is nuclear ((§1.7)[#section-1.7]).For every equicontinuous weakly closed disk

A inE' , there exists anotherB\supset A such that the identity map fromE'_A toE'_B is nuclear.

If

*integral* ((§1.8)[#section-1.8]).
Applying the same result to *Banach* space *weak* isomorphism.

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

Let

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

Let

Let

We do not know of any case where we have

Let

In particular, if

## 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:

—

- Let
E be an(\mathscr{F}) -space or a(\mathscr{DF}) -space. Then, forE to be nuclear, it is necessary and sufficient forE' to be nuclear. - Let
E be a nuclear space. Then every vector subspace and every quotient space ofE is nuclear. - 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.
- Let
E andF be two nuclear spaces. ThenE\hat{\otimes}F is nuclear. - Let
E be a nuclear space of type(\mathscr{F}) , andF a reflexive space whose dual is nuclear. Then the dual ofE\hat{\otimes}F is nuclear.

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

From Theorem 3, we obtain the following results:
if

Theorem 3, along with the fact that the space

All of the following are nuclear:
the spaces

*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

## 2.4 Lifting properties

[PTT, chap. 2, §3, no. 1]

Theorem 6 of §1.7 can be specialised as follows:

—

Let

E_1 andE_2 be locally convex spaces, andF_1 (resp.F_2 ) a vector subspace ofE_1 (resp.E_2 ). Suppose thatF_1 orF_2 is nuclear. Then every continuous bilinear form onF_1\times F_2 is the restriction of a nuclear bilinear form onE_1\times E_2 .Let

E andG be locally convex spaces, andF a vector subspace ofE . Suppose thatF is nuclear and thatG is quasi-complete, or thatG is the strong dual of a quasi-barrelled nuclear space (for example, thatG is a complete nuclear space of type(\mathscr{F}) or(\mathscr{DF}) ). Then every bounded linear map fromF toG is the restriction of a nuclear map fromE toG .Let

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

A variant of the above properties is the following:
Let

Let

The bounded disk

We note again that the analogue of Theorem 5 for the representation of *convergent sequences* in

## 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

Let

Subsequently, if

In general, if

The product

By the comments of §2.1, concretely determining

Let

Let

In particular, we find that

Let

We have analogous statements given by taking

## 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 *Fredholm kernels* to those which come from an element of some space * p-th power summable Fredholm kernel in E\bar{\otimes}F*.
(If

We define a * p-th power summable map* from

*not locally convex*. When

We can also, for *Fredholm kernels of order \leqslant p*, which is defined to be the intersection of the spaces

Let *rapidly decreasing* sequence of scalars, and where

Let *Fredholm determinant*

If *of genus 0*, and thus identical to the infinite product

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

If

The sequence of eigenvalues of a Fredholm operator of order

Furthermore, we can show that, under the conditions of Theorem 8, when

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

Let

In any case, the composition of

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

The combination of Theorem 1 and Theorem 9 gives:

Let

Let

Let

Other corollaries can be obtained by taking into account the other results given in this section.
Thus, if *rapidly decreasing* sequence.
If

## 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 *less fine* (and sometimes strictly less fine) than the topology induced by

E\hat{\otimes}F is bornological;E\hat{\otimes}F is barrelled;\mathrm{B}(E,F) is complete;\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 *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:*

The most important consequence of this criterion is that the space

The spaces

# Bibliography

*Algèbre multilinéaire*. Hermann, 1958.

*Actualites Scientifiques Et Industrielles*.

*Intégration*. Hermann, 1965.

*Actualites Scientifiques Et Industrielles*.

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**1**(1949), 61–101.

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**51**(1950), 387–408.

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**3**(1954), 57–123.

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*Actualites Scientifiques Et Industrielles*.

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.↩︎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.”*↩︎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*Since the spaceM be a locally compact and paracompact space (e.g. a compact space), andf a continuous map fromM to a quotient spaceE/F of an(\mathscr{F}) -spaceE . Thenf comes from a continuous map fromM toE .\mathscr{E}^{(m)}(V) ofm -times continuously differentiable functions on a infinitely differentiable paracompact manifoldV is isomorphic to a direct factor of a space of the formC(M) (as I noted in “Sur les applications linéaires faiblement compacts d’espaces du typeC(K) ,”*Can. J. Math.***5**(1953), p. 144), it thus follows that the analogous lifting theorem holds also form -times continuously differentiable maps fromV toE/F .↩︎