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\begin{document}

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\section*{nuclear space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{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 [[dual vector space|duality]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{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).

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

\end{defn}
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 set|convex]] [[circled set|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 \textbf{bounded} if for some $0$-neighbourhood $U$ of $E$ (which we may take to be [[circled set|circled]] and [[convex set|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 $f(U)$.

\begin{defn}
\label{nmap}\hypertarget{nmap}{}
Let $E$ and $F$ be LCTVS. A linear map $f \colon E \to F$ is \textbf{nuclear} if there exists a [[convex set|convex]] [[circled set|circled]] $0$-neighbourhood, say $U$, in $E$ and a [[convex set|convex]] [[circled set|circled]] [[bounded set|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.

\end{defn}
The following characterisation of nuclear maps is often helpful.

\begin{lemma}
\label{nchar}\hypertarget{nchar}{}
A linear map $f \colon E \to F$ is nuclear if and only if it is of the form:

\begin{displaymath}
u(x) = \sum_{n=1}^\infty \lambda_n f_n(x) y_n
\end{displaymath}
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.

\end{lemma}
Now that we have the notion of a nuclear map, we can define a nuclear space.

\begin{defn}
\label{nsp}\hypertarget{nsp}{}
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.

\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{enumerate}%
\item The following are equivalent:\begin{enumerate}%
\item $E$ is nuclear,
\item Every continuous linear map of $E$ into any Banach space is nuclear,
\item Every convex, circled $0$-neighbourhood $U$ contains another, say $V$, such that the canonical map $\tilde{E}_V \to \tilde{E}_U$ is nuclear.

\end{enumerate}

\item Every bounded subset of a nuclear space is precompact.
\item The completion of a nuclear space is a nuclear space.
\item A nuclear space is a projective limit of $\ell^p$ spaces (in particular, of [[Hilbert spaces]]).
\item 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.
\item The [[projective tensor product]] (and its completion) of two nuclear spaces is nuclear.

\end{enumerate}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{enumerate}%
\item Any finite dimensional vector space is nuclear,
\item For a finite dimensional smooth manifold $M$, $C^\infty(M)$ is nuclear,
\item The space of rapidly decreasing functions on $\mathbb{R}^n$ is nuclear,
\item The product an arbitrary number of copies of $\mathbb{R}$ is nuclear,
\item The direct sum of a countable number of copies of $\mathbb{R}$ is nuclear,
\item The direct sum of $\mathbb{R}$-copies of $\mathbb{R}$ is \emph{not} nuclear.

\end{enumerate}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item A. Grothendieck, \emph{Produits tensoriels topologiques et espaces nucl\'e{}aires}, Memoirs of the Amer. Math. Soc. \textbf{16}, 190 pp. and 140 pp. (1955).
\item [[Alexander Grothendieck]], \emph{R\'e{}sum\'e{} des r\'e{}sultats essentiels dans la th\'e{}orie des produits tensoriels topologiques et des espaces nucl\'e{}aires}, Annales de l'institut Fourier \textbf{4} (1952), p. 73-112, \href{http://www.numdam.org/item?id=AIF_1952__4__73_0}{numdam}

\end{itemize}
category: functional analysis

[[!redirects Nuclear space]] [[!redirects nuclear space]] [[!redirects nuclear spaces]] [[!redirects nuclear vector space]] [[!redirects nuclear vector spaces]] [[!redirects nuclear topological vector space]] [[!redirects nuclear topological vector spaces]]



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