Nonstandard topologies with bases that consist only of standard sets

Author:
Nader Vakil

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2075-2083

MSC (2000):
Primary 46S20; Secondary 54J05

DOI:
https://doi.org/10.1090/S0002-9939-00-05790-7

Published electronically:
December 4, 2000

MathSciNet review:
1825920

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Let be an infinite set, a set of pseudo-metrics on and If is limited (finite) for every and every then, for each we can define a pseudo-metric on by writing st We investigate the conditions under which the topology induced on by has a basis consisting only of standard sets. This investigation produces a theory with a variety of applications in functional analysis. For example, a specialization of some of our general results will yield such classical compactness theorems as Schauder's theorem, Mazur's theorem, and Gelfand-Philips's theorem.

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Additional Information

**Nader Vakil**

Affiliation:
Department of Mathematics, Western Illinois University, Macomb, Illinois 61455

Email:
N-Vakil@wiu.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05790-7

Keywords:
Nonstandard topologies,
Banach space,
compactness conditions

Received by editor(s):
June 8, 1999

Received by editor(s) in revised form:
November 16, 1999

Published electronically:
December 4, 2000

Communicated by:
Carl G. Jockusch, Jr.

Article copyright:
© Copyright 2000
American Mathematical Society