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Proceedings of the American Mathematical Society

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Pseudo-uniform convergence,
a nonstandard treatment


Author: Nader Vakil
Journal: Proc. Amer. Math. Soc. 126 (1998), 809-814
MSC (1991): Primary 46S20, 03H05
DOI: https://doi.org/10.1090/S0002-9939-98-04312-3
MathSciNet review: 1443413
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Abstract: We introduce and study the notion of pseudo-uniform convergence which is a weaker variant of quasi-uniform convergence. Applications include the following nonstandard characterization of weak convergence. Let $X$ be an infinite set, $B(X)$ the Banach space of all bounded real-valued functions on $X,$ $\{f_{n}: n\in N\}$ a bounded sequence in $B(X),$ and $f\in B(X).$ Then the sequence converges weakly to $f$ if and only if the convergence is pointwise on $X$ and, for each strictly increasing function $\sigma :N\to N$, each $x\in \ ^{*}X$, and each $n\in \ ^{*}N_{\infty }$, there is an unlimited $m\leq n$ such that $\ ^{*}f_{\ ^{*}\sigma (m)}(x)\ \simeq \ ^{*}f(x)$.


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Additional Information

Nader Vakil
Affiliation: Department of Mathematics, Western Illinois University, Macomb, Illinois 61455
Email: N-Vakil@bgu.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04312-3
Received by editor(s): October 31, 1995
Received by editor(s) in revised form: September 6, 1996
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1998 American Mathematical Society