F-space

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 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. 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 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. is an F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space. 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 The open mapping theorem implies that if are topologies on that make both and into complete metrizable topological vector spaces (for example, Banach or Fréchet spaces) and if one topology is finer or coarser than the other then they must be equal (that is, if ). A linear almost continuous map into an F-space whose graph is closed is continuous.