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F space

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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

Contents

  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

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

Clearly, 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).

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 1

L 1 2 [ 0 , 1 ] is an F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.

Example 2

Let W p ( D ) be the space of all complex valued Taylor series

f ( z ) = ∑ n ≥ 0 a n z n

on the unit disc D such that

∑ n | a n | p < ∞

then (for 0 < p < 1) W p ( D ) are F-spaces under the p-norm:

∥ f ∥ p = ∑ n | a n | p ( 0 < p < 1 )

In fact, W p is a quasi-Banach algebra. Moreover, for any ζ with | ζ | ≤ 1 the map f ↦ f ( ζ ) is a bounded linear (multiplicative functional) on W p ( D ) .

References

F-space Wikipedia
(Text) CC BY-SA


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