A collection of sequence spaces

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 \[ h_\alpha (x) = \operatorname {lub}_{p \in \mathcal {P}} \sum \limits _{i = 1}^\infty |x_{F(i)} \alpha _i| < k.\] $h_\alpha$ 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.