Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1970-0265913-8
MathSciNet review: 0265913
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] M. M. Day, Normed linear spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 21, Springer-Verlag, Berlin, 1958. MR 20 #1187. MR 0094675 (20:1187)
  • [2] D. J. H. Garling, On symmetric sequence spaces, Proc. London Math. Soc. (3) 16 (1966), 85-106. MR 33 #537. MR 0192311 (33:537)
  • [3] A. Pelczyński and W. Z. Szlenk, An example of a non-shrinking basis, Rev. Roumaine Math. Pures Appl. 10 (1965), 961-966. MR 34 #3284. MR 0203432 (34:3284)
  • [4] J. R. Retherford, A semishrinking basis which is not shrinking, Proc. Amer. Math. Soc. 19 (1968), 766. MR 37 #739. MR 0225144 (37:739)
  • [5] W. Ruckle, Symmetric coordinate spaces and symmetric bases, Canad. J. Math. 19 (1967), 828-838. MR 38 #4977. MR 0236682 (38:4977)
  • [6] W. L. C. Sargent, Some sequence spaces related to the $ {l^p}$ spaces, J. London Math. Soc. 35 (1960), 161-171. MR 22 #7001. MR 0116206 (22:7001)
  • [7] I. Singer, Sur les espaces de Banach à base absolut, canoniquement équivalents à un dual d'espace de Banach, C.R. Acad. Sci. Paris 251 (1960), 620-621. MR 22 #9841. MR 0115066 (22:5869)
  • [8] B. E. Wilder, Manifolds in a conjugate space, Ph.D. dissertation, Auburn University, Auburn, Alabama, 1966.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46.10

Retrieve articles in all journals with MSC: 46.10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0265913-8
Keywords: $ {l_p}$ spaces, sequence space, equivalent norms, Schauder basis, shrinking and semishrinking basis, norm attaining functionals
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society