Countably-normed space

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A locally convex space whose topology is defined using a countable set of compatible norms i.e. norms such that if a sequence that is fundamental in the norms and converges to zero in one of these norms, then it also converges to zero in the other. The sequence of norms can be replaced by a non-decreasing sequence , where , which generates the same topology with base of neighbourhoods of zero . A countably-normed space is metrizable, and its metric can be defined by

An example of a countably-normed space is the space of entire functions that are analytic in the unit disc with the topology of uniform convergence on any closed subset of this disc and with the collection of norms .


[1] I.M. Gel'fand, G.E. Shilov, "Generalized functions" , Acad. Press (1964) (Translated from Russian)
How to Cite This Entry:
Countably-normed space. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article