Fréchet space

A Fréchet space (or T1 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 $a \in U$, $b \notin U$ and $b \in V$, $a \notin V$.

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 $||\cdot||$, a Fréchet space is a complete topological vector space whose topology is defined by a countably infinite family of seminorms. For example, the space $C^\infty(\mathbb R)$ is Fréchet, with topological induced by the Ck norms

$||f||_{k,n} = \sup_{x \in [-n,n]} f^{(k)}(x)$.

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