Banach-Steinhaus Theorem

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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:

$\displaystyle \forall x \in X: \sup_{\alpha \mathop \in A} \norm {T_\alpha x}_Y < \infty$


Then:

$\displaystyle \sup_{\alpha \mathop \in A} \norm {T_\alpha} < \infty$

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



Proof


Also known as

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


Source of Name

This entry was named for Stefan Banach and Władysław Hugo Dionizy Steinhaus.


Sources