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Concepts of differentiability and analyticity on certain classes of topological groups

Published online by Cambridge University Press:  24 October 2008

G. A. Reid
Affiliation:
St John's College, Cambridge

Abstract

We introduce the concepts of a local seminorm on a topological group and of a locally convex group, showing that discrete groups, locally compact Abelian groups and compact groups are locally convex, and that a topological vector space is locally convex as a linear space if and only if it is locally convex as a group. We show that notions of differentiability, analyticity and derivability can be defined for locally convex groups and that these notions are suitably related and well behaved. We prove that for a locally compact Abelian group G the Fourier transforms of measures of compact support on the character group Ĝ are analytic, and for G compact the coefficients of continuous irreducible unitary representations are. Using these spaces of analytic functions we define the basic concepts of a differential geometry.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

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