The Banach-Steinhaus theorem (Uniform Boundedness Principle) states that for a family of bounded linear operators $\{T_\alpha\}_{\alpha \in A}$ from a Banach space $X$ to a normed space $Y$, pointwise boundedness implies uniform boundedness. Specifically, if
$$\sup_{\alpha \in A} \|T_\alpha(x)\| < \infty \quad \forall x \in X$$