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 metric, the two definitions are equivalent.

Proof

#missing/proof

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