# Banach-Steinhaus Theorem

## Theorem

### Normed Vector Space

Let $\struct {X, \norm {\,\cdot\,}_X}$ be a Banach space.

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

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:

- $\ds \forall x \in X: \sup_{\alpha \mathop \in A} \norm {T_\alpha x}_Y$ is finite.

Then:

- $\ds \sup_{\alpha \mathop \in A} \norm {T_\alpha}$ is finite

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

### Topological Vector Space

Let $\GF \in \set {\R, \C}$.

Let $X$ and $Y$ be topological vector spaces over $\GF$.

Let $\Gamma$ be a set of continuous linear transformations $X \to Y$.

Let $B$ be the set of all $x \in X$ such that:

- $\map \Gamma x = \set {T x : T \in \Gamma}$

is von Neumann-bounded in $Y$.

Suppose that $B$ is not meager in $X$.

Then $B = X$ and $\Gamma$ is equicontinuous.

### $F$-Space

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \tau_X}$ be an $F$-Space over $\GF$.

Let $\struct {Y, \tau_Y}$ be a topological vector space over $\GF$.

Let $\Gamma$ be a set of continuous linear transformations $X \to Y$ such that for all $x \in X$:

- $\map \Gamma x = \set {T x : T \in \Gamma}$ is von Neumann-bounded in $Y$.

Then $\Gamma$ is equicontinuous.

## Source of Name

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

## Historical Note

The Banach-Steinhaus Theorem was first proved, in the context of normed vector spaces, by Eduard Helly in around $1912$.

This was some years before Stefan Banach's work, but Helly failed to obtain recognition for this.