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

###### Definition

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

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 $\widetilde{E}_U$ as follows: as $U$ is convex and circled, its Minkowski functional is a semi-norm on $E$. The quotient $E_U \coloneqq E/\ker U$ 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 $\tilde{E}_U$ 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 $f \colon E \to F$ is bounded if for some $0$-neighbourhood $U$ of $E$ (which we may take to be circled and convex), $f(U)$ is bounded in $F$. In which case, $f$ factors through a continuous map $f_{U,B} \colon E_U \to F_B$ where $B \subseteq F$ is bounded and contains $f(U)$.

###### Definition

Let $E$ and $F$ be LCTVS. A linear map $f \colon E \to F$ 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 $f(U) \subseteq B$ and the associated map $f_{U,B} \colon \tilde{E}_U \to F_B$ is nuclear.

The following characterisation of nuclear maps is often helpful.

###### Lemma

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

$u(x) = \sum_{n=1}^\infty \lambda_n f_n(x) y_n$

where $\sum_{n=1}^\infty {|\lambda_n|} \lt \infty$, $\{f_n\}$ is an equicontinuous sequence in $E^*$, and $\{y_n\}$ 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 $\mathcal{B}$ of convex circled $0$-neighbourhoods such that for $V \in \mathcal{B}$ the canonical mapping $E \to \tilde{E}_V$ 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 $\tilde{E}_V \to \tilde{E}_U$ 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 $M$, $C^\infty(M)$ is nuclear,
3. The space of rapidly decreasing functions on $\mathbb{R}^n$ is nuclear,
4. The product an arbitrary number of copies of $\mathbb{R}$ is nuclear,
5. The direct sum of a countable number of copies of $\mathbb{R}$ is nuclear,
6. The direct sum of $\mathbb{R}$-copies of $\mathbb{R}$ is not nuclear.
• 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