Uniform boundedness principle, Uniform boundedness principle - Generalization, Uniform boundedness principle - Uniform boundedness principle, barrelled space, a topological vector space with minimum requirements for the Banach Steinhaus theorem to hold
ARTICLES RELATED TO Uniform boundedness principle - Generalization
The natural setting for the uniform boundedness principle is a barrelled space where the following generalized version of the theorem holds:
Given a barrelled space X and a locally convex space Y, then any family of pointwise bounded continuous linear mappings from X to Y is equicontinuous (even uniformly equicontinuous).
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More precisely, let X be a Banach space and N be a normed vector space. Suppose that F is a collection of continuous linear operators from X to N. The uniform boundedness principle states that if for all x in X we have
then
Using the Baire category theorem, we have the following short proof:
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