Uniform boundedness principle
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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 <math>X<math> be a Banach space and <math>N<math> be a normed vector space. Suppose that <math>F<math> is a collection of continuous linear operators from <math>X<math> to <math>N<math>. The uniform boundedness principle states that if for all x in X we have
- <math>\sup \left\{\,||T_\alpha (x)|| : T_\alpha \in F \,\right\} < \infty, <math>
then
- <math> \sup \left\{\, ||T_\alpha|| : T_\alpha \in F \;\right\} < \infty. <math>
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 (http://matwbn.icm.edu.pl/ksiazki/fm/fm09/fm0908.pdf)". Fundamenta Mathematicae, 9 50-61, 1927. (in french)he:משפט באנאך-שטיינהאוס