F-space

Template:For Template:Short description In functional analysis, an F-space is a vector space <math>X</math> over the real or complex numbers together with a metric <math>d : X \times X \to \R</math> such that

Scalar multiplication in <math>X</math> is continuous with respect to <math>d</math> and the standard metric on <math>\R</math> or <math>\Complex.</math>
Addition in <math>X</math> is continuous with respect to <math>d.</math>
The metric is translation-invariant; that is, <math>d(x + a, y + a) = d(x, y)</math> for all <math>x, y, a \in X.</math>
The metric space <math>(X, d)</math> is complete.

The operation <math>x \mapsto \|x\| := d(0, x)</math> is called an F-norm, although in general an F-norm is not required to be homogeneous. 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 Template:Em rather than Template:Em, 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 topological vector space. 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 <math>d(a x, 0) = |a| d(x, 0).</math><ref>Dunford N., Schwartz J.T. (1958). Linear operators. Part I: general theory. Interscience publishers, inc., New York. p. 59</ref>

The L^p spaces can be made into F-spaces for all <math>p \geq 0</math> and for <math>p \geq 1</math> they can be made into locally convex and thus Fréchet spaces and even Banach spaces.

Example 1Edit

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

Example 2Edit

Let <math>W_p(\mathbb{D})</math> be the space of all complex valued Taylor series <math display=block>f(z) = \sum_{n \geq 0} a_n z^n</math> on the unit disc <math>\mathbb{D}</math> such that <math display=block>\sum_n \left|a_n\right|^p < \infty</math> then for <math>0 < p < 1,</math> <math>W_p(\mathbb{D})</math> are F-spaces under the p-norm: <math display=block>\|f\|_p = \sum_n \left|a_n\right|^p \qquad (0 < p < 1).</math>

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

Sufficient conditionsEdit

Template:Math theorem

Related propertiesEdit

The open mapping theorem implies that if <math>\tau \text{ and } \tau_2</math> are topologies on <math>X</math> that make both <math>(X, \tau)</math> and <math>\left(X, \tau_2\right)</math> into complete metrizable topological vector spaces (for example, Banach or Fréchet spaces) and if one topology is finer or coarser than the other then they must be equal (that is, if <math>\tau \subseteq \tau_2 \text{ or } \tau_2 \subseteq \tau \text{ then } \tau = \tau_2</math>).Template:Sfn

A linear almost continuous map into an F-space whose graph is closed is continuous.Template:Sfn
A linear almost open map into an F-space whose graph is closed is necessarily an open map.Template:Sfn
A linear continuous almost open map from an F-space is necessarily an open map.Template:Sfn
A linear continuous almost open map from an F-space whose image is of the second category in the codomain is necessarily a surjective open map.Template:Sfn