F-space
In functional analysis, an F-space 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 translation-invariant; 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.
The operation x ↦ ||x|| := d(0,x) is called an F-norm, although in general an F-norm is not required to be complete. 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 call these spaces Fréchet spaces, but usually the term is reserved for locally convex F-spaces. 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 d(αx, 0) = |α|⋅d(x, 0).^{[1]}
The L^{p} spaces are F-spaces for all p ≥ 0 and for p ≥ 1 they are 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 0 < p < 1) 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 .
See also
References
- ^ 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, McGraw-Hill, ISBN 0-07-054234-1