Pointwise bounded families of operators on a Banach space are uniformly bounded in operator norm.Suggest a slogan
Let \(X\) be a Banach space, let \(Y\) be a normed vector space, and let \(\mathcal{F}\) be a family of bounded linear operators \(T\colon X\to Y\). If \[ \sup_{T \in \mathcal{F}} \norm{Tx}_Y < \infty \quad\text{for every } x\in X, \] then \(\sup_{T\in \mathcal{F}} \norm{T} < \infty\), where \(\norm{T}\coloneqq\sup_{\norm{x}_X\le1}\norm{Tx}_Y\) is the operator norm.
State the uniform boundedness principle.