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. Together with the Hahn-Banach theorem and the open mapping theorem, it is 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.

Contents 1 Uniform boundedness principle 2 Generalization 3 See also 4 References

[edit] Uniform boundedness principle

The precise statement of the result is:

Theorem 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 X_n = { x : ||T(x)|| ≤ n (∀ T ∈ F) } . By hypothesis, the union of all the X_n is X.

Since X is a Baire space, one of the X_n has an interior point, giving some δ > 0 and y in X such that ||x - y|| < δ ⇒ x ∈ X_n.

For any x such that ||x|| ≤ δ, ||T(x)|| ≤ ||T(y - x)|| + ||T(y)|| ≤ n + n = 2n

Hence for all T ∈ F, ||T|| < 2n/δ, so that 2n/δ is a uniform bound for the set F.

A direct consequence is:

Theorem If a sequence of bounded operators {T_n} converges pointwise, that is, lim T_n(x) exists for all x in X, then these pointwise limits define a bounded operator T.

The convergence T_n → T is only uniform on compact subsets; the sequence may fail to converge in operator norm.

[edit] 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).

[edit] See also

• barrelled space, a topological vector space with minimum requirements for the Banach Steinhaus theorem to hold

[edit] References

• Stefan Banach, Hugo Steinhaus. "Sur le principle de la condensation de singularités". Fundamenta Mathematicae, 9 50-61, 1927. (in french)