nLab nuclear space

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.

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

category: functional analysis