F-space

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For F-spaces in general topology, see sub-Stonean space.

In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d : V × VR 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

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[edit]

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).[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 1[edit]

\scriptstyle L^\frac{1}{2}[0,\, 1] is a F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.

Example 2[edit]

Let \scriptstyle W_p(\mathbb{D}) be the space of all complex valued Taylor series

f(z)=\sum_{n \geq 0}a_n z^n

on the unit disc \scriptstyle \mathbb{D} such that

\sum_{n}|a_n|^p < \infty

then (for 0 < p < 1) \scriptstyle W_p(\mathbb{D}) are F-spaces under the p-norm:

\|f\|_p= \sum_{n}|a_n|^p \qquad (0 < p < 1)

In fact, \scriptstyle W_p is a quasi-Banach algebra. Moreover, for any \scriptstyle \zeta with \scriptstyle |\zeta| \;\leq\; 1 the map \scriptstyle f \,\mapsto\, f(\zeta) is a bounded linear (multiplicative functional) on \scriptstyle W_p(\mathbb{D}).

References[edit]

  1. ^ Dunford N., Schwartz J.T. (1958). Linear operators. Part I: general theory. Interscience publishers, inc., New York. p. 59