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Pointwise convergence

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2020 Mathematics Subject Classification: Primary: 54C35 [MSN][ZBL]

A type of convergence of sequences of functions (mappings). Let , n=1,2,\ldots where X is some set and Y is a topological space; then pointwise convergence means that for any element x \in X the sequence of values y_n = f_n(x), n=1,2,\ldots converges in the space Y. The function f : x \mapsto \lim_n y_n is then the pointwise limit of the sequence (f_n). The definition extends to generalized sequences of functions and their values.

An important subclass of the pointwise-convergent sequences for the case of mappings between metric spaces (or, more generally, uniform spaces) is that of the uniformly-convergent sequences (cf. Uniform convergence).

See also Pointwise convergence, topology of.

How to Cite This Entry:
Pointwise convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_convergence&oldid=40133
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article