Fréchet space

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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.


[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)



[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:
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article