In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
is defined to be a locally convex metrizable topological vector space (TVS) that is complete as a TVS,[1] meaning that every Cauchy sequence in
[note 1] The topology of every Fréchet space is induced by some translation-invariant complete metric.
Conversely, if the topology of a locally convex space
[1] The local convexity requirement was added later by Nicolas Bourbaki.
[1] It's important to note that a sizable number of authors (e.g. Schaefer) use "F-space" to mean a (locally convex) Fréchet space while others do not require that a "Fréchet space" be locally convex.
Moreover, some authors even use "F-space" and "Fréchet space" interchangeably.
When reading mathematical literature, it is recommended that a reader always check whether the book's or article's definition of "F-space" and "Fréchet space" requires local convexity.
[1] Fréchet spaces can be defined in two equivalent ways: the first employs a translation-invariant metric, the second a countable family of seminorms.
is a Fréchet space if and only if it satisfies the following three properties: Note there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology.
The alternative and somewhat more practical definition is the following: a topological vector space
is a Fréchet space if and only if it satisfies the following three properties: A family
in the Fréchet space defined by a family of seminorms if and only if it converges to
Theorem[3] (de Wilde 1978) — A topological vector space
In contrast to Banach spaces, the complete translation-invariant metric need not arise from a norm.
Even though the topological structure of Fréchet spaces is more complicated than that of Banach spaces due to the potential lack of a norm, many important results in functional analysis, like the open mapping theorem, the closed graph theorem, and the Banach–Steinhaus theorem, still hold.
with the following two properties: Then the topology induced by these seminorms (as explained above) turns
A translation-invariant complete metric inducing the same topology on
Consequently, a product of uncountably many non-trivial Fréchet spaces can not be a Fréchet space (indeed, such a product is not even metrizable because its origin can not have a countable neighborhood basis).
into complete metrizable TVSs (such as Fréchet spaces) and if one topology is finer or coarser than the other then they must be equal (that is, if
[9] The strong dual of a DF-space is a Fréchet space.
is a Fréchet space on which there does not exist any continuous norm, then this is due entirely to the presence of
is a non-normable Fréchet space on which there exists a continuous norm, then
contains a closed vector subspace that has no topological complement.
is infinite dimensional) then its strong dual space
is normable if (and only if) there exists a complete norm on its continuous dual space
Anderson–Kadec theorem — Every infinite-dimensional, separable real Fréchet space is homeomorphic to
the Cartesian product of countably many copies of the real line
This is in stark contrast to the situation in Banach spaces.
LF-spaces are countable inductive limits of Fréchet spaces.