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 in Wikipedia the term Fréchet space is reserved for locally convex F-spaces.
Examples
Clearly, all Banach spaces and Fréchet spaces are F-spaces.
The Lp spaces are F-spaces for all p>0 and for p ≥ 1 they are even locally convex and thus Fréchet spaces. So for example, L1/2[0,1] is a F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.