Please use this identifier to cite or link to this item:
http://hdl.handle.net/11375/5856
Title: | Topological Algebras with Orthogonal M-bases |
Authors: | El-Helaly, Taha Sherif |
Advisor: | Husain, T. |
Department: | Mathematics |
Keywords: | Mathematics;Mathematics |
Publication Date: | Aug-1985 |
Abstract: | <p>An M-basis in a topological vector space is a special case of the extended Markushevich basis, and a generalization of the unconditional basis. We study orthogonal bases and orthogonal M-bases in topological algebras, with emphasis on locally convex algebras. It turns out that an orthogonal basis or an orthogonal M-basis in a topological algebra is necessarily Schauder. We characterize some concrete topological algebras with orthogonal bases or orthogonal M-bases, up to a topological isomorphism. We introduce and study two classes of locally convex algebras: the class of "Φ-algebras" which includes, for example ℂʳ, c₀(r), C*₃(r) and H(D) (with the Hadamard multiplication); and the larger class of "locally convex s-algebras" which also includes - among other examples - ℓp, 1 ≤ p < ∞ and the Arens algebra L^ω[0,1]. A Φ-algebras is not necessarily locally m-convex, and a locally m-convex algebra is not necessarily a locally convex s-algebra. We give two examples of Banach algebras with orthogonal bases which are not unconditional and we prove that an orthogonal basis in a B₀ algebra is unconditional iff the algebra is a locally convex s-algebra. We also study the conversion of a Fréchet space with an unconditional basis into a Fréchet algebra with the basis under consideration as an orthogonal basis and we obtain a necessary and sufficient condition for this to be possible, revising and extending a result of Husain and Watson obtained for Banach spaces.</p> |
URI: | http://hdl.handle.net/11375/5856 |
Identifier: | opendissertations/1202 2498 1304001 |
Appears in Collections: | Open Access Dissertations and Theses |
Files in This Item:
File | Size | Format | |
---|---|---|---|
fulltext.pdf | 2.06 MB | Adobe PDF | View/Open |
Items in MacSphere are protected by copyright, with all rights reserved, unless otherwise indicated.