Advanced | Help | Encyclopedia
Directory


Uniform boundedness principle

(Redirected from Banach-Steinhaus theorem)

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.

Table of contents

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).

See also

References








Links: Addme | Keyword Research | Paid Inclusion | Femail | Software | Completive Intelligence

Add URL | About Slider | FREE Slider Toolbar - Simply Amazing
Copyright © 2000-2008 Slider.com. All rights reserved.
Content is distributed under the GNU Free Documentation License.