View source for F-space

{{for|F-spaces in general topology|sub-Stonean space}}
{{short description|Topological vector space with a complete translation-invariant metric}}
In [[functional analysis]], an '''F-space''' is a [[vector space]] <math>X</math> over the [[Real number|real]] or [[Complex number|complex]] numbers together with a [[Metric (mathematics)|metric]] <math>d : X \times X \to \R</math> such that
# Scalar multiplication in <math>X</math> is [[Continuous function#Continuous functions between metric spaces|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 metric|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 metric space|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|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 {{em|[[Fréchet space]]}} rather than {{em|F-space}}, but usually the term "Fréchet space" is reserved for [[Locally convex topological vector space|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 topological vector space|metrizable]] in a manner that satisfies the above properties.

== Examples ==

All [[Banach space]]s and [[Fréchet space]]s 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 [[Lp space|L<sup>p</sup> 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 1 ===

<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 2 ===

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 conditions ==

{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|p=35}}<ref name="Klee Inv metrics">{{Cite journal|last1=Klee|first1=V. L.|title=Invariant metrics in groups (solution of a problem of Banach)|year=1952|journal=Proc. Amer. Math. Soc.|volume=3|issue=3|pages=484–487|url=https://www.ams.org/journals/proc/1952-003-03/S0002-9939-1952-0047250-4/S0002-9939-1952-0047250-4.pdf|doi = 10.1090/s0002-9939-1952-0047250-4|doi-access=free}}</ref>|note=Klee (1952)|math_statement=
Let <math>d</math> be {{em|any}}<ref group=note>Not assume to be translation-invariant.</ref> metric on a vector space <math>X</math> such that the topology <math>\tau</math> induced by <math>d</math> on <math>X</math> makes <math>(X, \tau)</math> into a topological vector space. If <math>(X, d)</math> is a complete metric space then <math>(X, \tau)</math> is a [[complete topological vector space]].
}}

== Related properties ==

The [[Open mapping theorem (functional analysis)|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 topological vector space|complete]] [[metrizable topological vector space]]s (for example, Banach or [[Fréchet space]]s) and if one topology is [[Comparison of topologies|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>).{{sfn|Trèves|2006|pp=166–173}}

* A linear [[almost continuous]] map into an F-space whose graph is closed is continuous.{{sfn|Husain|Khaleelulla|1978|p=14}}
* A linear [[Almost open map|almost open]] map into an F-space whose graph is closed is necessarily an [[open map]].{{sfn|Husain|Khaleelulla|1978|p=14}}
* A linear continuous [[Almost open map|almost open]] map from an F-space is necessarily an [[open map]].{{sfn|Husain|Khaleelulla|1978|p=15}}
* A linear continuous almost open map from an F-space whose image is of the [[second category]] in the codomain is necessarily a [[Surjection|surjective]] [[open map]].{{sfn|Husain|Khaleelulla|1978|p=14}}

== See also ==

* {{annotated link|Banach space}}
* {{annotated link|Barreled space}}
* {{annotated link|Countably quasi-barrelled space}}
* {{annotated link|Complete metric space}}
* {{annotated link|Complete topological vector space}}
* {{annotated link|DF-space}}
* {{annotated link|Fréchet space}}
* {{annotated link|Hilbert space}}
* {{annotated link|K-space (functional analysis)}}
* {{annotated link|LB-space}}
* {{annotated link|LF-space}}
* {{annotated link|Metrizable topological vector space}}
* {{annotated link|Nuclear space}}
* {{annotated link|Projective tensor product}}

== References ==

{{reflist}}

==Notes==
{{reflist|group=note}}

==Sources==
{{sfn whitelist |CITEREFHusainKhaleelulla1978}}
* {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}} <!-- {{sfn|Husain|Khaleelulla|1978|p=}} -->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|1982|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Cite book|last=Rudin|first=Walter|author-link=Walter Rudin|title=Real & Complex Analysis|publisher=McGraw-Hill|year=1966|isbn=0-07-054234-1}}
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Schechter Handbook of Analysis and Its Foundations}} <!-- {{sfn|Schechter|1996|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->

{{Functional Analysis}}
{{TopologicalVectorSpaces}}

[[Category:F-spaces| ]]
[[Category:Topological vector spaces]]

[[pl:Przestrzeń Frécheta (analiza funkcjonalna)]]