F-spaces

F-space

In functional analysis, an F-space is a vector space over the real or complex numbers together with a metric such that 1. * Scalar multiplication in is continuous with respect to and the standard metric on or 2. * Addition in is continuous with respect to 3. * The metric is translation-invariant; that is, for all 4. * The metric space is complete. The operation is called an F-norm, although in general an F-norm is not required to be homogeneous. By translation-invariance, the metric is recoverable from the F-norm. Thus, a real or complex F-space is equivalently a real or complex vector space equipped with a complete F-norm. Some authors use the term Fréchet space rather than F-space, but usually the term "Fréchet space" is reserved for locally convex F-spaces. Some other authors use the term "F-space" as a synonym of "Fréchet space", by which they mean a locally convex complete metrizable topological vector space. The metric may or may not necessarily be part of the structure on an F-space; many authors only require that such a space be metrizable in a manner that satisfies the above properties. (Wikipedia).

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Comparison of topologies | Locally convex topological vector space | Metrizable topological vector space | Almost open map | Banach space | Complex number | Fréchet space | Functional analysis | Vector space | Lp space | Dual space | Open mapping theorem (functional analysis) | Real number | Complete topological vector space | Complete metric space | Continuous function | Taylor series | Topological vector space