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, anal i.e. there exists 𝑟 >0 such that 𝑑(𝑥,𝑦) <𝑟 for all 𝑥,𝑦 ∈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

proof

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