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 <math>a \in U</math>, <math>b \notin U</math> and <math>b \in V</math>, <math>a \notin V</math>.

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

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

Note that these are only seminorms, and not honest norms, since <math>||f||_{k,n}</math> may be 0 even if <math>f</math> is not.