# Fréchet space

A complete metrizable locally convex topological vector space. Banach spaces furnish examples of Fréchet spaces, but several important function spaces are Fréchet spaces without being Banach spaces. Among these are: the Schwartz space of all infinitely-differentiable complex-valued functions on that decrease at infinity, as do all their derivatives, faster than any polynomial, with the topology given by the system of semi-norms

where and are non-negative integer vectors; the space of all holomorphic functions on some open subset of the complex plane with the topology of uniform convergence on compact subsets of , etc.

A closed subspace of a Fréchet space is a Fréchet space; so is a quotient space of a Fréchet space by a closed subspace; a Fréchet space is a barrelled space, and therefore the Banach–Steinhaus theorem is true for mappings from a Fréchet space into a locally convex space. If a separable locally convex space is the image of a Fréchet space under an open mapping, then it is itself a Fréchet space. A one-to-one continuous linear mapping from a Fréchet space onto a Fréchet space is an isomorphism (an analogue of a theorem of Banach).

Fréchet spaces are so named in honour of M. Fréchet.

#### References

[1] | N. Bourbaki, "Topological vector spaces" , Springer (1987) (Translated from French) |

[2] | A.P. Robertson, W.S. Robertson, "Topological vector spaces" , Cambridge Univ. Press (1973) |

#### Comments

#### References

[a1] | H.H. Schaefer, "Topological vector spaces" , Macmillan (1966) |

[a2] | J.L. Kelley, I. Namioka, "Linear topological spaces" , Springer (1963) |

[a3] | G. Köthe, "Topological vector spaces" , 1 , Springer (1969) |

**How to Cite This Entry:**

Fréchet space.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_space&oldid=14937