nuclear space

If $E$ is a Fréchet space and $(p_j)$ an increasing sequence of semi-norms on $E$ defining the topology of $E$, we have

$$E=\underset{⟵}{lim}{\widehat{E}}_{p_j},$$

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 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 $f:E\to F$ 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 ,${\xi }_j\in E^{\prime }$ an equicontinuous sequence of linear forms and $y_j\in F$ a bounded sequence.