# Fréchet space

A **Fréchet space** (or **T _{1} spaces**) is a topological space

*X*in wich for any two points

*a*,

*b*, there exist a pair of open sets

*U*and

*V*, such that , and , .

More commonly, the term **Fréchet space** is applied to an unrelated object in functional analysis. It is a natural generalization of the notion of a Banach space, and the majority of theorems about Banach spaces equally well apply to Fréchet spaces. Whereas a Banach space is a complete topological vector space with the topology induced by some norm , a Fréchet space is a complete topological vector space whose topology is defined by a countably infinite family of seminorms. For example, the space is Fréchet, with topological induced by the norms

- .

Note that these are only seminorms, and not honest norms, since may be 0 even if is not.