# nuclear space

If $E$ is a FrÃÂ©chet 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 FrÃÂ©chet 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 $f\colon E\to F$ 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 $\sum|\lambda_{j}|<+\infty$,$\xi_{j}\in E^{\prime}$ an equicontinuous sequence of linear forms and $y_{j}\in F$ a bounded sequence.

Title nuclear space NuclearSpace 2013-03-22 16:37:28 2013-03-22 16:37:28 Simone (5904) Simone (5904) 6 Simone (5904) Definition msc 46B20