Pointwise convergence
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.
Pointwise convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_convergence&oldid=40133