Banach-Steinhaus Theorem/Normed Vector Space

Theorem
Let $X$ be a Banach space.

Let $Y$ be a normed vector space with norm $\norm {\,\cdot\,}_Y$.

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

Suppose that:
 * $\ds \forall x \in X: \sup_{\alpha \mathop \in A} \norm {T_\alpha x}_Y < \infty$

Then:
 * $\ds \sup_{\alpha \mathop \in A} \norm {T_\alpha} < \infty$

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

Also known as
This theorem is also known as the uniform boundedness principle or uniform bounded principle.