Theorem 12.1.6 (Uniform Boundedness Principle).label Let $E, F$ be normed vector spaces and $\mathcal{T}\subset L(E; F)$. If
- (B1)
$E$ is a Banach space.
- (B2)
For every $x \in E$, $\sup_{T \in \mathcal{T}}\norm{Tx}_{F} < \infty$.
then $\sup_{T \in \mathcal{T}}\norm{T}_{L(E; F)}< \infty$.
Proof. By the Banach-Steinhaus Theorem, $\mathcal{T}$ is equicontinuous. Therefore there exists $r > 0$ such that $\bigcup_{T \in \mathcal{T}}T[B_{E}(0, r)] \subset B_{F}(0, 1)$. In which case, $\sup_{T \in \mathcal{T}}\norm{T}_{L(E; F)}\le r^{-1}$.$\square$
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