# 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

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.

The operation x ↦ ||x|| := d(0,x) is called an F-norm, although in general an F-norm is not required to be complete. By translation-invariance, the metric is recoverable from the F-norm. Thus, a real or complex F-space is equivalently a real or complex vector space equipped with a complete F-norm.

Some authors use the term Fréchet space rather than F-space, but usually the term "Fréchet space" is reserved for locally convex F-spaces. Some other authors use the term "F-space" as a synonym of "Fréchet space", by which they mean a locally convex complete metrizable TVSs. 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

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).[1]

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

${\displaystyle \scriptstyle L^{\frac {1}{2}}[0,\,1]}$  is an F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.

### Example 2

Let ${\displaystyle \scriptstyle W_{p}(\mathbb {D} )}$  be the space of all complex valued Taylor series

${\displaystyle f(z)=\sum _{n\geq 0}a_{n}z^{n}}$

on the unit disc ${\displaystyle \scriptstyle \mathbb {D} }$  such that

${\displaystyle \sum _{n}|a_{n}|^{p}<\infty }$

then (for 0 < p < 1) ${\displaystyle \scriptstyle W_{p}(\mathbb {D} )}$  are F-spaces under the p-norm:

${\displaystyle \|f\|_{p}=\sum _{n}|a_{n}|^{p}\qquad (0

In fact, ${\displaystyle \scriptstyle W_{p}}$  is a quasi-Banach algebra. Moreover, for any ${\displaystyle \scriptstyle \zeta }$  with ${\displaystyle \scriptstyle |\zeta |\;\leq \;1}$  the map ${\displaystyle \scriptstyle f\,\mapsto \,f(\zeta )}$  is a bounded linear (multiplicative functional) on ${\displaystyle \scriptstyle W_{p}(\mathbb {D} )}$ .

## Sufficient conditions

Theorem[2][3] (Klee) — Let d be any[note 1] metric on a vector space X such that the topology 𝜏 induced by d on X makes (X, 𝜏) into a topological vector space. If (X, d) is a complete metric space then (X, 𝜏) is a complete-TVS.