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
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- 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.
Examples
Clearly, all Banach spaces and Fréchet spaces are F-spaces. In particular, a Banach space is an F-space with an additional requirement that d(αx, 0) = |α|⋅d(x, 0).
The Lp spaces are F-spaces for all p ≥ 0 and for p ≥ 1 they are locally convex and thus Fréchet spaces and even Banach spaces.
Example 1
Example 2
Let
on the unit disc
then (for 0 < p < 1)
In fact,