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Generating classes of perfect Banach sequence spaces


Author: G. Crofts
Journal: Proc. Amer. Math. Soc. 36 (1972), 137-143
MSC: Primary 46A45
DOI: https://doi.org/10.1090/S0002-9939-1972-0312208-7
MathSciNet review: 0312208
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Abstract: A perfect sequence space $ \lambda $ is said to be a step if $ {l^1} \subset \lambda \subset {l^\infty }$ and $ \lambda $ is a Banach space in its strong topology from $ {\lambda ^{\text{x}}}$. In this paper a method is given to generate additional steps from a step $ \lambda $. Precisely, $ {\lambda ^p}$ is a step where $ {\lambda ^p} \equiv \{ x = ({x_i})\vert{x_i} \in C$ and $ \vert x{\vert^p} = (\vert{x_i}{\vert^p}) \in \lambda \} $, for $ 1 \leqq p < \infty $, with norm $ {\left\Vert x \right\Vert _{{\lambda ^p}}} = {({\left\Vert {\vert x{\vert^p}} \right\Vert _\lambda })^{1/p}}$. It is shown that $ {\lambda ^p}$, $ 1 < p < \infty $, is reflexive iff $ \lambda $ has a Schauder basis. The space of diagonal maps of $ {\lambda ^p}$ into $ \lambda $ is characterized, as is the space of diagonal nuclear maps of $ \lambda $ into $ {\lambda ^p}$ when $ \lambda $ has a Schauder basis.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0312208-7
Keywords: Perfect sequence space, Banach space, regular sequence space, step, diagonal maps, multipliers of sequence spaces, nuclear maps, diagonal nuclear maps, reflexive
Article copyright: © Copyright 1972 American Mathematical Society

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