Uniform boundedness principle

In mathematics, the uniform boundedness principle (sometimes known as the Banach-Steinhaus Theorem) is one of the fundamental results of functional analysis. 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.

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>

In some texts, one finds this called the Banach-Steinhaus Theorem, since it is a generalisation of a theorem first appearing in a 1927 paper of Stefan Banach and Hugo Dyonizy Steinhaus. The uniform boundedness principle is often considered one of the three cornerstone theorems of functional analysis, the others being the Hahn-Banach theorem and the open mapping theorem.

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 such that ||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.

A version of the uniform boundedness principle also holds for F-spaces, with uniform boundedness being replaced by uniform equicontinuity.