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

  1. Scalar multiplication in V is continuous with respect to d and the standard metric on ℝ or ℂ.
  2. Addition in V is continuous with respect to d.
  3. The metric is translation-invariant; i.e., d(x + a, y + a) = d(x, y) for all x, y and a in V
  4. 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 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 TVSs. 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.

ExamplesEdit

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 Lp 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 1Edit

  is an F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.

Example 2Edit

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  .

Sufficient conditionsEdit

Theorem[2][3] (Klee) — Let d be any[note 1] metric on a vector space X such that the topology 𝜏 induced by d on X makes (X, 𝜏) into a topological vector space. If (X, d) is a complete metric space then (X, 𝜏) is a complete-TVS.

Related propertiesEdit

See alsoEdit

ReferencesEdit

  1. ^ Not assume to be translation-invariant.
  1. ^ Dunford N., Schwartz J.T. (1958). Linear operators. Part I: general theory. Interscience publishers, inc., New York. p. 59
  2. ^ Schaefer & Wolff 1999, p. 35.
  3. ^ Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4.
  4. ^ a b c Husain 1978, p. 14.
  5. ^ Husain 1978, p. 15.