Let $(X,\lVert \cdot \rVert_{X}),(Y,\lVert \cdot \rVert_{Y})$ be Banach spaces. Let $(\phi_{\alpha})_{\alpha\in\Lambda}$ be a family of continuous linear mappings s.t. $\phi_{\alpha}:X\to Y$. Assume $\forall x\in X: \sup_{\alpha\in\Lambda}\lVert \phi_{\alpha}(x) \rVert_{Y}<\infty$. Then $$\sup_{\alpha\in\Lambda}\lVert \phi_{\alpha} \rVert _{\text{op}}<\infty$$where $$\lVert \phi_{\alpha} \rVert _{\text{op}}=\sup_{\lVert x \rVert _{X}\le 1}\lVert \phi_{\alpha}(x) \rVert _{Y}$$