F-space (original) (raw)

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

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

Some authors call these spaces "Fréchet spaces", but in Wikipedia the term Fréchet space is reserved for locally convex F-spaces.

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.