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nuclear space (Definition)

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

$\displaystyle 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 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

$\displaystyle f(x)=\sum\lambda_j\xi_j(x)y_j $
where $ (\lambda_j)$ is a sequence of scalars with $ \sum\vert\lambda_j\vert<+\infty$, $ \xi_j\in E'$ an equicontinuous sequence of linear forms and $ y_j\in F$ a bounded sequence.



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"nuclear space" is owned by Simone. [ full author list (2) ]
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Keywords:  Fréchet space, Banach space, semi-norm
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Cross-references: bounded, linear forms, equicontinuous, scalars, convex, complete, nuclear, induced norm, Banach space, morphism, canonical, completion, Hausdorff, topology, semi-norms, sequence, increasing
There are 4 references to this entry.

This is version 3 of nuclear space, born on 2007-01-26, modified 2007-01-26.
Object id is 8823, canonical name is NuclearSpace.
Accessed 630 times total.

Classification:
AMS MSC: 46B20 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Geometry and structure of normed linear spaces)
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