Universal generators for varieties of nuclear spaces

Rosenberger, B.

doi:10.1090/S0002-9947-1973-0328522-0

Universal generators for varieties of nuclear spaces
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Trans. Amer. Math. Soc. 184 (1973), 275-290 Request permission

Abstract:

It is shown that a product of several copies of $\Lambda ({\beta ^\phi })$ is a universal $\phi$-nuclear space if the power series space $\Lambda ({\beta ^\phi })$ with $\beta _k^\phi = - \log ({\phi ^{ - 1}}(1/\sqrt {k + 1} )),k\;\epsilon \;\{ 0,1,2, \cdots \}$, is $\phi$-nuclear; here $\phi = [0,\infty ) \to [0,\infty )$ is a continuous, strictly increasing subadditive function with $\phi (0) = 0$. In case $\Lambda ({\beta ^\phi })$ is not $\phi$-nuclear the sequence space $\Lambda (l_\phi ^ + )$ is a $\phi$-nuclear space with the property that every $\phi$-nuclear space is isomorphic to a subspace of a product of $\Lambda (l_\phi ^ + )$ if ${\lim \;\sup _{t \to 0}}{(\phi (t))^{ - 1}}\phi (\sqrt t ) < \infty$.

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