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 (∀ TF) } . 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 ⇒ xXn.
Hence for all TF, ||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).

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