Fréchet space

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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.

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