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 = d for all x, y and a in V
  4. The metric space is complete.
The operation x ↦ ||x|| := d 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.
The Lp spaces are F-spaces for all and for they are locally convex and thus Fréchet spaces and even Banach spaces.

Example 1

is an F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.

Example 2

Let be the space of all complex valued Taylor series
on the unit disc such that
then are F-spaces under the p-norm:
In fact, is a quasi-Banach algebra. Moreover, for any with the map is a bounded linear on.

Related properties