[[Analysis MOC]]
# Bounded set

A **bounded set** $S$ is a set in a [[metric space]] $(M,d)$ for which all points are within a certain distance of each other, #m/def/anal 
i.e. there exists $r > 0$ such that $d(x,y) < r$ for all $x,y \in S$.

## Properties

- [[Compact sets in a metric space are bounded]]

## Topological vector space

For a topological vector space, there exists a concept of [[Von Neumann bounded set]].
If the topology is induced by an [[Absolutely homogenous function|absolutely homogenous]] metric,
the two definitions are equivalent.

> [!missing]- Proof
> #missing/proof


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#state/tidy | #lang/en | #SemBr