F-space

For F-spaces in general topology, see sub-Stonean 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 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

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 L^p 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

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

Example 2

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

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

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

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

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

$\|f\|_{p}=\sum _{n}|a_{n}|^{p}\qquad (0<p<1)$

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

See also

K-space (functional analysis)

References

• Rudin, Walter (1966), Real & Complex Analysis, McGraw-Hill, ISBN 0-07-054234-1