# Contents

## Idea

A nuclear vector space is a locally convex topological vector space that is as far from being a normed vector space as possible. Any map from a nuclear space into a normed vector space is compact, whence the only normed nuclear spaces are finite dimensional.

Nuclear spaces have very good properties with regard to topological tensor product?s and duality.

## Definition

To define a nuclear space we need to start with the concept of a nuclear map, first between Banach spaces.

Let $E$ and $F$ be Banach spaces. Let $ℒ\left(E,F\right)$ be the Banach space of continuous linear maps $E\to F$. Let ${E}^{*}$ denote the dual Banach space of $E$. Let ${E}^{*}\stackrel{˜}{\otimes }F$ denote the completion of the projective tensor product of ${E}^{*}$ and $F$. The bilinear map ${E}^{*}×F\to ℒ\left(E,F\right)$ extends to a continuous linear map ${E}^{*}\stackrel{˜}{\otimes }F\to ℒ\left(E,F\right)$ (which might not be injective).

###### Definition

Let $E$ and $F$ be Banach spaces. A linear map $f:E\to F$ is nuclear if it lies in the image in $ℒ\left(E,F\right)$ of the completion of the projective tensor product ${E}^{*}\stackrel{˜}{\otimes }F$.

From the notion of nuclear maps between Banach spaces we can define nuclear maps between arbitrary LCTVS. In essence, a linear map between arbitrary LCTVS is nuclear if it factors through a nuclear map of Banach spaces.

To make this precise, we need to recall how to associate Banach spaces to certain subsets of an LCTVS. Let $E$ be an LCTVS and $U\subseteq E$ a convex circled $0$-neighbourhood. Then we can define a Banach space ${\stackrel{˜}{E}}_{U}$ as follows: as $U$ is convex and circled, its Minkowski functional is a semi-norm on $E$. The quotient ${E}_{U}≔E/\mathrm{ker}U$ is therefore a normed vector space. As $U$ is a $0$-neighbourhood, the quotient mapping defines a continuous linear function $E\to {E}_{U}$. We define ${\stackrel{˜}{E}}_{U}$ to be the Banach completion of ${E}_{U}$.

There is a dual notion. Let $F$ be an LCTVS and $B\subseteq F$ a convex, circled, and bounded subset of $F$. Let ${F}_{B}$ be the span of $B$ in $F$. Then $B$ is absorbing in ${F}_{B}$ and so its Minkowski functional is defined. If $F$ is Hausdorff then $B$ cannot contain a linear subspace and thus ${F}_{B}$ is a normed vector space. We cannot complete ${F}_{B}$ to a Banach space but it might so happen that it is one. As $B$ is bounded, the inclusion ${F}_{B}\to F$ is continuous.

(There is no danger of confusing the two notations since if $E$ admits a bounded $0$-neighbourhood then it is a normed vector space.)

Now we say that a continuous linear map $f:E\to F$ is bounded if for some $0$-neighbourhood $U$ of $E$ (which we may take to be circled and convex), $f\left(U\right)$ is bounded in $F$. In which case, $f$ factors through a continuous map ${f}_{U,B}:{E}_{U}\to {F}_{B}$ where $B\subseteq F$ is bounded and contains $f\left(U\right)$.

###### Definition

Let $E$ and $F$ be LCTVS. A linear map $f:E\to F$ is nuclear if there exists a convex circled $0$-neighbourhood, say $U$, in $E$ and a convex circled bounded, say $B$, in $F$ with ${F}_{B}$ complete such that $f\left(U\right)\subseteq B$ and the associated map ${f}_{U,B}:{\stackrel{˜}{E}}_{U}\to {F}_{B}$ is nuclear.

The following characterisation of nuclear maps is often helpful.

###### Lemma

A linear map $f:E\to F$ is nuclear if and only if it is of the form:

$u\left(x\right)=\sum _{n=1}^{\infty }{\lambda }_{n}{f}_{n}\left(x\right){y}_{n}$u(x) = \sum_{n=1}^\infty \lambda_n f_n(x) y_n

where ${\sum }_{n=1}^{\infty }\mid {\lambda }_{n}\mid <\infty$, $\left\{{f}_{n}\right\}$ is an equicontinuous sequence in ${E}^{*}$, and $\left\{{y}_{n}\right\}$ is a sequence in $F$ contained in a convex, circled, bounded subset $B$ such that ${F}_{B}$ is complete.

Now that we have the notion of a nuclear map, we can define a nuclear space.

###### Definition

A LCTVS $E$ is nuclear if it has a base $ℬ$ of convex circled $0$-neighbourhoods such that for $V\in ℬ$ the canonical mapping $E\to {\stackrel{˜}{E}}_{V}$ is nuclear.

## Properties

1. The following are equivalent:

1. $E$ is nuclear,
2. Every continuous linear map of $E$ into any Banach space is nuclear,
3. Every convex, circled $0$-neighbourhood $U$ contains another, say $V$, such that the canonical map ${\stackrel{˜}{E}}_{V}\to {\stackrel{˜}{E}}_{U}$ is nuclear.
2. Every bounded subset of a nuclear space is precompact.

3. The completion of a nuclear space is a nuclear space.

4. A nuclear space is a projective limit of ${\ell }^{p}$ spaces (in particular, of Hilbert spaces).

5. Nuclearity is inherited by the following constructions: subspaces, separated quotients, arbitrary products, locally convex direct sum of a countable family, projective limits, countable inductive limits.

6. The projective tensor product (and its completion) of two nuclear spaces is nuclear.

## Examples

1. Any finite dimensional vector space is nuclear,
2. For a finite dimensional smooth manifold $M$, ${C}^{\infty }\left(M\right)$ is nuclear,
3. The space of rapidly decreasing functions on ${ℝ}^{n}$ is nuclear,
4. The product an arbitrary number of copies of $ℝ$ is nuclear,
5. The direct sum of a countable number of copies of $ℝ$ is nuclear,
6. The direct sum of $ℝ$-copies of $ℝ$ is not nuclear.

## References

• A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Memoirs of the Amer. Math. Soc. 16, 190 pp. and 140 pp. (1955).
• Alexander Grothendieck, Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires, Annales de l’institut Fourier 4 (1952), p. 73-112, numdam

Revised on June 9, 2013 12:31:35 by Toby Bartels (173.190.139.188)