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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Universal generators for varieties of nuclear spaces


Author: B. Rosenberger
Journal: Trans. Amer. Math. Soc. 184 (1973), 275-290
MSC: Primary 46A05
MathSciNet review: 0328522
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0328522-0
PII: S 0002-9947(1973)0328522-0
Keywords: Nuclear spaces, sequence spaces, variety, universal generator
Article copyright: © Copyright 1973 American Mathematical Society