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Topological and algebraic genericity and spaceability for an extended chain of sequence spaces

Topological and algebraic genericity and spaceability for an extended chain of sequence spaces Given a pair of topological vector spaces X,Y\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X,\; Y$$\end{document} where X is a proper linear subspace of Y it is examined whether Y\X\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Y\setminus X$$\end{document} is residual in Y (topological genericity), whether Y\X\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Y\setminus X$$\end{document} contains a dense linear subspace of Y except 0 (algebraic genericity) and whether Y\X\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Y\setminus X$$\end{document} contains a closed infinite dimensional subspace of Y except 0 (spaceability). In the present paper the spaces X and Y are either sequence spaces or spaces of analytic functions on the unit disc regarded as sequence spaces via the identification of a function with the sequence of its Taylor coefficients. For the spaces under consideration we give an affirmative answer to each of these questions providing general proofs which extend previous results. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte für Mathematik Springer Journals

Topological and algebraic genericity and spaceability for an extended chain of sequence spaces

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References (13)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2022
ISSN
0026-9255
eISSN
1436-5081
DOI
10.1007/s00605-022-01732-y
Publisher site
See Article on Publisher Site

Abstract

Given a pair of topological vector spaces X,Y\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X,\; Y$$\end{document} where X is a proper linear subspace of Y it is examined whether Y\X\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Y\setminus X$$\end{document} is residual in Y (topological genericity), whether Y\X\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Y\setminus X$$\end{document} contains a dense linear subspace of Y except 0 (algebraic genericity) and whether Y\X\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Y\setminus X$$\end{document} contains a closed infinite dimensional subspace of Y except 0 (spaceability). In the present paper the spaces X and Y are either sequence spaces or spaces of analytic functions on the unit disc regarded as sequence spaces via the identification of a function with the sequence of its Taylor coefficients. For the spaces under consideration we give an affirmative answer to each of these questions providing general proofs which extend previous results.

Journal

Monatshefte für Mathematik – Springer Journals

Published: Mar 1, 2023

Keywords: Topological genericity; Algebraic genericiy; Spaceability; Baire’s theorem; ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^p$$\end{document} spaces; Topological genericity; Algebraic genericiy; Spaceability; Baire’s theorem; ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^p$$\end{document} spaces; 46A45; 46E10; 46E15

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