Uniform boundedness principle

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, $Y$ a normed vector space and $B(X,Y)$ the space of all continuous linear operators from $X$ into $Y$ . Suppose that $F$ is a collection of continuous linear operators from $X$ to $Y.$ If, for every $x\in X$ ,

\sup _{T\in F}\|T(x)\|_{Y}<\infty ,

then

\sup _{T\in F}\|T\|_{B(X,Y)}=\sup _{\stackrel {T\in F,}{\|x\|\leq 1}}\|T(x)\|_{Y}<\infty .

In the case that

X

is not the trivial vector space, then the semi-inequality used in the supremum of the first term in this last chain of equalities (which has

x

range over the closed unit ball) may be replaced by a proper equality (which has

x

range over the closed unit sphere).

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

Proof

Let $X$ be a Banach space. Suppose that for every $x\in X,$

\sup _{T\in F}\|T(x)\|_{Y}<\infty .

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

X_{n}=\left\{x\in X\ :\ \sup _{T\in F}\|T(x)\|_{Y}\leq n\right\}.

Each set $X_{n}$ is a closed set and by the assumption,

\bigcup _{n\in \mathbb {N} }X_{n}=X\neq \varnothing .

By the Baire category theorem for the non-empty complete metric space $X,$ there exists some $m\in \mathbb {N}$ such that $X_{m}$ has non-empty interior; that is, there exist $x_{0}\in X_{m}$ and $\varepsilon >0$ such that

{\overline {B_{\varepsilon }(x_{0})}}~:=~\left\{x\in X\,:\,\|x-x_{0}\|\leq \varepsilon \right\}~\subseteq ~X_{m}.

Let $u\in X$ with $\|u\|\leq 1$ and $T\in F.$ Then:

{\begin{aligned}\|T(u)\|_{Y}&=\varepsilon ^{-1}\left\|T\left(x_{0}+\varepsilon u\right)-T\left(x_{0}\right)\right\|_{Y}&\\&\leq \varepsilon ^{-1}\left(\left\|T(x_{0}+\varepsilon u)\right\|_{Y}+\left\|T(x_{0})\right\|_{Y}\right)\\&\leq \varepsilon ^{-1}(m+m).&\\\end{aligned}}