# 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.

## Historical Note

The Banach-Steinhaus Theorem was first proved by Eduard Helly in around $1912$, some years before Stefan Banach's work, but he failed to obtain recognition for this.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
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