NuclearSpace 2007-01-26 09:53:40 2007-01-26 12:14:29 Definition Fr\'echet space Banach space semi-norm % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here If $E$ is a Fréchet space and $(p_j)$ an increasing sequence of semi-norms on $E$ defining the topology of $E$, we have $$ E=\underset{\longleftarrow}{\lim}\,\widehat E_{p_j}, $$ where $\widehat E_{p_j}$ is the Hausdorff completion of $(E,p_j)$ and $\widehat E_{p_{j+1}}\to\widehat E_{p_j}$ the canonical morphism. Here $\widehat E_{p_j}$ is a Banach space for the induced norm $\widehat p_j$. A Fréchet space $E$ is said to be \emph{nuclear} if the topology of $E$ can be defined by an increasing sequence of semi-norms $p_j$ such that each canonical morphism $\widehat E_{p_{j+1}}\to\widehat E_{p_j}$ of Banach spaces is nuclear. Recall that a morphism $f\colon E\to F$ of complete locally convex spaces is said to be nuclear if $f$ can be written as $$ f(x)=\sum\lambda_j\xi_j(x)y_j $$ where $(\lambda_j)$ is a sequence of scalars with $\sum|\lambda_j|<+\infty$,$\xi_j\in E'$ an equicontinuous sequence of linear forms and $y_j\in F$ a bounded sequence.