Pseudo-uniform convergence, a nonstandard treatment
HTML articles powered by AMS MathViewer
- by Nader Vakil PDF
- Proc. Amer. Math. Soc. 126 (1998), 809-814 Request permission
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 $^{*}f_{ ^{*}\sigma (m)}(x) \simeq ^{*}f(x)$.References
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- John W. Brace, Compactness in function spaces, Duke Math. J. 29 (1962), 157–166. MR 187212
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
- Albert E. Hurd and Peter A. Loeb, An introduction to nonstandard real analysis, Pure and Applied Mathematics, vol. 118, Academic Press, Inc., Orlando, FL, 1985. MR 806135
- W. A. J. Luxemburg, A general theory of monads, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 18–86. MR 0244931
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Additional Information
- Nader Vakil
- Affiliation: Department of Mathematics, Western Illinois University, Macomb, Illinois 61455
- Email: N-Vakil@bgu.edu
- Received by editor(s): October 31, 1995
- Received by editor(s) in revised form: September 6, 1996
- Communicated by: Andreas R. Blass
- © Copyright 1998 American Mathematical Society
- 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