The operation is called an F-norm, although in general an F-norm is not required to be homogeneous. 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 topological vector space.
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 [1]
The Lp spaces can be made into F-spaces for all and for they can be made into 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 for are F-spaces under the p-norm:
In fact, is a quasi-Banach algebra. Moreover, for any with the map is a bounded linear (multiplicative functional) on
Sufficient conditions
Theorem[2][3](Klee (1952)) — Let be any[note 1] metric on a vector space such that the topology induced by on makes into a topological vector space. If is a complete metric space then is a complete topological vector space.