##### Wikipedia

**F-space**

In functional analysis, an **F-space** is a vector space*V* over the real or complex numbers together with a metric*d* : *V* × *V* → **R** so that

- Scalar multiplication in
*V*is continuous with respect to*d*and the standard metric on**R**or**C**. - Addition in
*V*is continuous with respect to*d*. - The metric is translation-invariant; i.e.,
*d*(*x*+*a*,*y*+*a*) =*d*(*x*,*y*) for all*x*,*y*and*a*in*V* - The metric space (
*V*,*d*) is complete

Some authors call these spaces * Fréchet spaces*, but usually the term is reserved for locally convex F-spaces. 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.