# Definition:Von Neumann-Bounded Subset of Topological Vector Space

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## Definition

Let $\Bbb F \in \set {\R, \C}$.

Let $\struct {X, \tau}$ be a topological vector space over $\Bbb F$.

Let $E \subseteq X$.

We say that $E$ is **von Neumann-bounded** if and only if:

- for each open neighbourhood $V$ of ${\mathbf 0}_X$, there exists $s > 0$ such that $E \subseteq t V$ for each $t > s$

where $t V$ is the dilation of $V$ by $t$.

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## Also see

## Source of Name

This entry was named for John von Neumann.

## Sources

- 1991: Walter Rudin:
*Functional Analysis*(2nd ed.) ... (previous) ... (next): $1.6$: Topological vector spaces