Banach-Steinhaus Theorem/Normed Vector Space

Thorem
Let $X$ be a Banach space.

Let $Y$ be a normed vector space with norm $\left\Vert{\cdot}\right\Vert_Y$.

Let $\left\langle{T_\alpha: X \to Y}\right\rangle_{\alpha \in A}$ be an $A$-indexed family of bounded linear transformations from $X$ to $Y$.

Suppose that:
 * $\displaystyle \forall x \in X: \sup_{\alpha \mathop \in A} \left\Vert{T_\alpha x}\right\Vert_Y < \infty$

Then:
 * $\displaystyle \sup_{\alpha \mathop \in A} \left\Vert{T_\alpha}\right\Vert < \infty$

where $\left\Vert{T_\alpha}\right\Vert$ denotes the norm of the linear transformation $T_\alpha$.

Also known as
This theorem is also known as the Uniform Boundedness Principle.