Uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach-Steinhaus Theorem is one of the fundamental results in functional analysis and, together with the Hahn-Banach theorem and the open mapping theorem, considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators whose domain is a Banach space, pointwise boundedness is equivalent to boundedness.
The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus but it was also proven independently by Hans Hahn.
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Uniform boundedness principle
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:
- For n = 1,2,3, ... let Xn = { x : ||T(x)|| ≤ n (∀ T ∈ F) } . By hypothesis, the union of all the Xn is X.
- Since X is a Baire space, one of the Xn has an interior point, giving some δ > 0 such that ||x|| < δ ⇒ x ∈ Xn.
- Hence for all T ∈ F, ||T|| < n/δ, so that n/δ is a uniform bound for the set F.
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).
See also
- barrelled space, a topological vector space with minimum requirements for the Banach Steinhaus theorem to hold
References
- Stefan Banach, Hugo Steinhaus. "Sur le principle de la condensation de singularités". Fundamenta Mathematicae, 9 50-61, 1927. (in french)