Birkhoff-Kakutani Theorem

Theorem

Topological Vector Space

Let $\GF \in \set {\R, \C}$.

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

Then $\struct {X, \tau}$ is pseudometrizable if and only if $\struct {X, \tau}$ is first-countable.

Further, if $\struct {X, \tau}$ is pseudometrizable then there exists an invariant pseudometric $d$ on $X$ such that:

$(1): \quad$ $d$ induces $\tau$

$(2): \quad$ the open balls in $\struct {X, d}$ are balanced.

Source of Name

This entry was named for Garrett Birkhoff and Shizuo Kakutani.