Pointwise convergence
A type of convergence of sequences of functions (mappings). Let ,
where
is some set and
is a topological space; then pointwise convergence means that for any element
the sequence of points
,
converges in the space
. 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).
Comments
A base for the topology of pointwise convergence on , the space of continuous mappings from
to
, is obtained as follows. Take a finite set
and for each
an open subset in
containing
; for a given
an open basis neighbourhood is:
. See also Pointwise convergence, topology of.
References
[a1] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 86 (Translated from Russian) |
Pointwise convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_convergence&oldid=35248