- 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.