Fréchet space

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 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 "-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. A topological vector space is a Fréchet space if and only if it satisfies the following three properties: It is locally convex.