Фреше кеңістіктері: Жергілікті дөңес топологиялық векторлық кеңістіктер

A locally convex topological vector space that is also a complete metric space
Fréchet spaces in functional analysis

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces. A Fréchet space is defined to be a locally convex metrizable topological vector space (TVS) that is complete as a TVS, meaning that every Cauchy sequence in converges to some point in (see footnote for more details). Important note: Not all authors require that a Fréchet space be locally convex (discussed below). The topology of every Fréchet space is induced by some translation invariant complete metric. Conversely, if the topology of a locally convex space is induced by a translation invariant complete metric then is a Fréchet space. Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space" to mean a complete metrizable topological vector space, without the local convexity requirement (such a space is today often called an "F space"). The local convexity requirement was added later by Nicolas Bourbaki. 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.

Fréchet spaces can be defined in two equivalent ways: the first employs a translation invariant metric, the second a countable family of seminorms.

In contrast to Banach spaces, the complete translation invariant metric need not arise from a norm. The topology of a Fréchet space does, however, arise from both a total paranorm and an F norm (the F stands for Fréchet). 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.

Every Banach space is a Fréchet space, as the norm induces a translation invariant metric and the space is complete with respect to this metric. The space of all real valued sequences (also denoted ) becomes a Fréchet space if we define the th seminorm of a sequence to be the absolute value of the th element of the sequence. Convergence in this Fréchet space is equivalent to element wise convergence.

Note that the homeomorphism described in the Anderson–Kadec theorem is not necessarily linear.

If we drop the requirement for the space to be locally convex, we obtain F spaces: vector spaces with complete translation invariant metrics. LF spaces are countable inductive limits of Fréchet spaces.