Wikipedia / F-space
Wikipedia / F-space
In functional analysis, an F-space is a vector space over the real or complex numbers together with a metric such that
is continuous
The metric is translation-invariant
is complete.
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 rather than , 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
The Lp spaces 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
be the space of all complex valued Taylor series are F-spaces under the p-norm
is a quasi-Banach algebra
Sufficient conditions
Related properties
The open mapping theorem into complete metrizable topological vector spaces (for example, Banach or Fréchet spaces) and if one topology is finer or coarser).
A linear almost continuous map into an F-space whose graph is closed is continuous.
A linear almost open map into an F-space whose graph is closed is necessarily an open map.
A linear continuous almost open map from an F-space is necessarily an open map.
A linear continuous almost open map from an F-space whose image is of the second category in the codomain is necessarily a surjective open map.
See also
References
Notes
Sources