Editing F-space (section)

{{for|F-spaces in general topology|sub-Stonean space}}
{{short description|Topological vector space with a complete translation-invariant metric}}
In [[functional analysis]], an '''F-space''' is a [[vector space]] <math>X</math> over the [[Real number|real]] or [[Complex number|complex]] numbers together with a [[Metric (mathematics)|metric]] <math>d : X \times X \to \R</math> such that
# Scalar multiplication in <math>X</math> is [[Continuous function#Continuous functions between metric spaces|continuous]] with respect to <math>d</math> and the standard metric on <math>\R</math> or <math>\Complex.</math>
# Addition in <math>X</math> is continuous with respect to <math>d.</math>
# The metric is [[Translation-invariant metric|translation-invariant]]; that is, <math>d(x + a, y + a) = d(x, y)</math> for all <math>x, y, a \in X.</math>
# The metric space <math>(X, d)</math> is [[Complete metric space|complete]].

The operation <math>x \mapsto \|x\| := d(0, x)</math> is called an '''F-norm''', although in general an F-norm is not required to be homogeneous. By [[Translation invariance|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 {{em|[[Fréchet space]]}} rather than {{em|F-space}}, but usually the term "Fréchet space" is reserved for [[Locally convex topological vector space|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 topological vector space|metrizable]] in a manner that satisfies the above properties.