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Pseudo-uniform convergence, a nonstandard treatment

Authors:	Nader Vakil

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

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 be an infinite set, the Banach space of all bounded real-valued functions on $\{f_{n}: n\in N\}$ a bounded sequence in and $f\in B(X).$ Then the sequence converges weakly to if and only if the convergence is pointwise on 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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