Uniform boundedness principle

Let \(X\) be a Banach space, let \(Y\) be a normed vector space, and let \((T_\alpha)_{\alpha\in A}\) be a family of bounded linear operators \(T_\alpha\colon X\to Y\). Suppose the family is pointwise bounded: for every \(x\in X\) there exists \(C_x\geq0\) such that \[ \norm{T_{\alpha}x} \leq C_x \quad\text{for all } \alpha\in A . \] Then the family is uniformly bounded in operator norm: there exists \(C\geq0\) such that \[ \norm{T_\alpha x}\leq C\norm{x} \quad\text{for all } \alpha\in A \text{ and all } x\in X, \] that is, \(\sup_{\alpha\in A}\norm{T_\alpha}<\infty\) [@Fol13].