von Neumann bounded set

Summary

A subset of a topological vector space is von Neumann bounded if a large enough scaling shrinks it into any given neighbourhood of $0$. It replaces the notion of a bounded set of a metric space. That notion does not suit topological vector spaces, since a metric may make the whole space bounded. In a normed space the von Neumann bounded sets are the sets of finite norm diameter. Bounded sets are used to define the strong dual space, bornological spaces and Montel spaces.