Fréchet space

2020 Mathematics Subject Classification: Primary: 46A04 [MSN][ZBL]

A Fréchet space is 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 $\mathscr{S}(\R^n)$ of all infinitely-differentiable complex-valued functions on $\R^n$ that decrease at infinity, as do all their derivatives, faster than any polynomial, with the topology given by the system of semi-norms \[ p_{\alpha,\beta}(x) = \sup_{t \in \R^n} \left| t_1^{\beta_1} \cdots t_n^{\beta_n} \frac{ \partial^{\alpha_1 + \cdots + \alpha_n} x(t_1,\ldots,t_n) }{ \partial t_1^{\alpha_1} \cdots \partial t_n^{\alpha_n} } \right|, \] where $\alpha$ and $\beta$ are non-negative integer vectors; the space $H(D)$ of all holomorphic functions on some open subset $D$ of the complex plane with the topology of uniform convergence on compact subsets of $D$, 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

[Bo]	N. Bourbaki, "Topological vector spaces", Springer (1987) (Translated from French)
[KeNa]	J.L. Kelley, I. Namioka, "Linear topological spaces", Springer (1963)
[Kö]	G. Köthe, "Topological vector spaces", 1, Springer (1969)
[RoRo]	A.P. Robertson, W.S. Robertson, "Topological vector spaces", Cambridge Univ. Press (1973)
[Sc]	H.H. Schaefer, "Topological vector spaces", Macmillan (1966)