- 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

Clearly, all Banach spaces and Fréchet spaces are F-spaces.
The L^{p} spacess for 0 < *p* < 1 are examples of F-spaces which are not Fréchet spaces.