F-space
Topological vector space with a complete translation-invariant metric / From Wikipedia, the free encyclopedia
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In functional analysis, an F-space is a vector space over the real or complex numbers together with a metric
such that
- Scalar multiplication in
is continuous with respect to
and the standard metric on
or
- Addition in
is continuous with respect to
- The metric is translation-invariant; that is,
for all
- The metric space
is complete.
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.