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

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

Examples

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 L^p 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

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

Example 2

Let $\scriptstyle W_{p}(\mathbb {D} )$ 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 $\scriptstyle W_{p}(\mathbb {D} )$ .

Related properties

A linear almost continuous map into an F-space whose graph is closed is continuous.[2]
A linear almost open map into an F-space whose graph is closed is necessarily an open map.[2]
A linear continuous almost open map from an F-space is necessarily an open map.[3]
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.[2]

References

Dunford N., Schwartz J.T. (1958). Linear operators. Part I: general theory. Interscience publishers, inc., New York. p. 59
Husain 1978, p. 14.
Husain 1978, p. 15.

Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.
Khaleelulla, S. M. (1982). Counterexamples in topological vector spaces. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
Rudin, Walter (1966), Real & Complex Analysis, McGraw-Hill, ISBN 0-07-054234-1

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

[FOOTNOTEHusain197814-2] Husain 1978, p. 14.

[FOOTNOTEHusain197815-3] Husain 1978, p. 15.

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

Topological vector spaces (TVSs)
Basic concepts	Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space
Main results	Closed graph theorem Hahn–Banach (hyperplane separation) Open mapping (Banach–Schauder) Uniform boundedness (Banach–Steinhaus)
Maps	Almost open Bilinear (form operator) and Sesquilinear forms Closed Compact operator Continuous and Discontinuous linear maps Densely defined Homomorphism Functionals Operator Seminorm Sublinear Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled (Ultra-) Bornological Brauner (DF)-space Distinguished F-space Fréchet (tame Fréchet) Grothendieck Hilbert Infrabarreled Interpolation space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the Approximation property