F-space

In functional analysis, an F-space is a vector space ${\displaystyle X}$ over the real or complex numbers together with a metric ${\displaystyle d:X\times X\to \mathbb {R} }$ such that

1. Scalar multiplication in ${\displaystyle X}$ is continuous with respect to ${\displaystyle d}$ and the standard metric on ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} .}$
2. Addition in ${\displaystyle X}$ is continuous with respect to ${\displaystyle d.}$
3. The metric is translation-invariant; that is, ${\displaystyle d(x+a,y+a)=d(x,y)}$ for all ${\displaystyle x,y,a\in X.}$
4. The metric space ${\displaystyle (X,d)}$ is complete.

The operation ${\displaystyle x\mapsto \|x\|:=d(0,x)}$ 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.