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An open mapping theorem

Published online by Cambridge University Press:  24 October 2008

Kung-Fu Ng
Kung-Fu Ng
Affiliation:
United College, The Chinese University of Hong Kong
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Extract

Let (E, τ) be a complete, semi-metrizable topological vector space. Let p be a pseudo-norm (not to be confused with a semi-norm, cf. (8), p. 18) inducing the topology τ. For each positive real number r, let

Let f be a continuous linear function from E into a topological vector space F. The open mapping theorem of Banach may be stated as follows: If f is nearly open, that is, if the closure of each f(Vr) is a neighbourhood of O in F then whenever β > α > O; in particular, each f(Vr) is a neighbourhood of O. We note that f, identifying with its graph, is a closed linear subspace of the product space E × F. In this paper, we shall employ techniques developed by Kelley (6) and Baker (1) to extend the theorem to the case where f is taken to be a closed cone in E × F. The generalized theorem throws some light onto the duality theory of ordered spaces. In particular, the theorem of Andô–Ellis is generalized to (not assumed, a priori to be complete) normed vector spaces.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 1 , July 1973 , pp. 61 - 66
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

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