Fréchet Space

A Fréchet space is a complete and metrizable space, sometimes also with the restriction that the space be locally convex. The topology of a Fréchet space is defined by a countable family of seminorms. For example, the space of smooth functions on  is a Fréchet space. Its topology is the C-infty topology, which is given by the countable family of seminorms,

Because  in this topology implies that  is smooth, i.e.,

any Cauchy sequence has a limit in the space of smooth functions, i.e., it is a complete vector space.

REFERENCES:

Schaefer, H. H. Topological Vector Spaces. New York: Macmillan, 1966.