 Uniform boundedness principle

The Uniform boundedness principle says that a pointwise bounded family of bounded linear operators from a Banach space $X$ to a normed vector space $Y$ has uniformly bounded operator norms. Being bounded pointwise means ${\sup }_T\Vert Tx\Vert <\infty$ for every fixed $x$; the conclusion is the stronger common bound ${\sup }_T\Vert T\Vert <\infty$.

For the proof, the closed sets $E_n=\{x:{\sup }_T\Vert Tx\Vert \le n\}$ cover $X$. The Baire category theorem makes one contain a ball. Subtracting two points in that ball bounds every linear operator on a ball about zero; rescaling gives the common operator norm bound. Completeness is the hypothesis that permits the Baire step. The principle turns individual scalar or vector bounds into a uniform estimate without having to compute each operator norm separately.

The Uniform Boundedness Principle, also known as the Banach-Steinhaus theorem, is a fundamental result in functional analysis. It provides conditions under which a family of bounded linear operators is uniformly bounded.