# 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.

## 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*X*= {_{n}*x*: ||*T*(*x*)|| ≤*n*(∀*T*∈*F*) } . By hypothesis, the union of all the*X*is_{n}*X*. - Since
*X*is a Baire space, one of the*X*has an interior point, giving some δ > 0 such that ||_{n}*x*|| < δ ⇒*x*∈*X*._{n} - 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)

Categories: Functional analysis | Theorems