Fspace
In functional analysis, an Fspace is a vector space V over the real or complex numbers together with a metric d : V × V → R so that
 Scalar multiplication in V is continuous with respect to d and the standard metric on R or C.
 Addition in V is continuous with respect to d.
 The metric is translationinvariant; i.e., d(x + a, y + a) = d(x, y) for all x, y and a in V
 The metric space (V, d) is complete
Some authors call these spaces Fréchet spaces, but usually the term is reserved for locally convex Fspaces. The metric may or may not necessarily be part of the structure on an Fspace; many authors only require that such a space be metrizable in a manner that satisfies the above properties.
Contents
Examples[edit]
Clearly, all Banach spaces and Fréchet spaces are Fspaces. In particular, a Banach space is an Fspace with an additional requirement that d(αx, 0) = α⋅d(x, 0).^{[1]}
The L^{p} spaces are Fspaces for all p > 0 and for p ≥ 1 they are locally convex and thus Fréchet spaces and even Banach spaces.
Example 1[edit]
is a Fspace. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.
Example 2[edit]
Let be the space of all complex valued Taylor series
on the unit disc such that
then (for 0 < p < 1) are Fspaces under the pnorm:
In fact, is a quasiBanach algebra. Moreover, for any with the map is a bounded linear (multiplicative functional) on .
References[edit]
 ^ Dunford N., Schwartz J.T. (1958). Linear operators. Part I: general theory. Interscience publishers, inc., New York. p. 59
 Rudin, Walter (1966), Real & Complex Analysis, McGrawHill, ISBN 0070542341
