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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A collection of sequence spaces

Authors: J. R. Calder and J. B. Hill
Journal: Trans. Amer. Math. Soc. 152 (1970), 107-118
MSC: Primary 46.10
MathSciNet review: 0265913
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Abstract: This paper concerns a collection of sequence spaces we shall refer to as $ {d_\alpha }$ spaces. Suppose $ \alpha = ({\alpha _1},{\alpha _2}, \ldots )$ is a bounded number sequence and $ {\alpha _i} \ne 0$ for some $ i$. Suppose $ \mathcal{P}$ is the collection of permutations on the positive integers. Then $ {d_\alpha }$ denotes the set to which the number sequence $ x = ({x_1},{x_2}, \ldots )$ belongs if and only if there exists a number $ k > 0$ such that

$\displaystyle h_\alpha(x) = \operatorname{lub}_{p \in \mathcal{P}} \sum\limits_{i = 1}^\infty \vert x_{F(i)} \alpha_i\vert < k.$

$ h_\alpha$ is a norm on $ d_\alpha$ and $ (d_\alpha, h_\alpha)$ is complete.

We classify the $ {d_\alpha }$ spaces and compare them with $ {l_1}$ and $ m$. Some of the $ {d_\alpha }$ spaces are shown to have a semishrinking basis that is not shrinking. Further investigation of the bases in these spaces yields theorems concerning the conjugate space properties of $ {d_\alpha }$. We characterize the sequences $ \beta $ such that, given $ \alpha ,{d_\beta }, = {d_\alpha }$. A class of manifolds in the first conjugate space of $ {d_\alpha }$ is examined. We establish some properties of the collection of points in the first conjugate space of a normed linear space $ S$ that attain their maximum on the unit ball in $ S$. The effect of renorming $ {c_0}$ and $ {l_1}$ with $ {h_\alpha }$ and related norms is studied in terms of the change induced on this collection of functionals.

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Keywords: $ {l_p}$ spaces, sequence space, equivalent norms, Schauder basis, shrinking and semishrinking basis, norm attaining functionals
Article copyright: © Copyright 1970 American Mathematical Society