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). Fréchet spaces are locally convex spaces that are complete with respect to a translation-invariant metric. In contrast to Banach spaces, the metric need not arise from a norm. 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). Fréchet spaces are locally convex spaces that are complete with respect to a translation-invariant metric. In contrast to Banach spaces, the 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 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. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces. Fréchet spaces can be defined in two equivalent ways: the first employs a translation-invariant metric, the second a countable family of semi-norms. A topological vector space X is a Fréchet space if and only if it satisfies the following three properties: