Banach–Steinhaus theorem

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 (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus but it was also proven independently by Hans Hahn.

Theorem (Uniform Boundedness Principle). Let X be a Banach space and Y be a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y. If for all x in X one has

then

The completeness of X enables the following short proof, using the Baire category theorem.

Proof. Suppose that for every x in the Banach space X, one has:

For every integer ${\displaystyle n\in \mathbb {N} }$, let

The set ${\displaystyle X_{n}}$ is a closed set and by the assumption,

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