F-space
In functional analysis, an F-space is a vector spaceover the real or complex numbers together with a metricis 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.All Banach spaces and Fréchet spaces are F-spaces.In particular, a Banach space is an F-space with an additional requirement that[1] The Lp spaces can be made into F-spaces for allthey can be made into locally convex and thus Fréchet spaces and even Banach spaces.It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.be the space of all complex valued Taylor serieson the unit discare F-spaces under the p-norm:is a quasi-Banach algebra.is a bounded linear (multiplicative functional) onbe any[note 1] metric on a vector spaceinto a topological vector space.is a complete metric space thenis a complete topological vector space.The open mapping theorem implies that ifinto complete metrizable topological vector spaces (for example, Banach or Fréchet spaces) and if one topology is finer or coarser than the other then they must be equal (that is, if