F-space

In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric $d : V \times V \to ℝ$ 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 $d (αx, 0) = |α|\cdot d (x, 0)$ .^[1]

The L^p spaces can be made into F-spaces for all p ≥ 0 and for p ≥ 1 they can be made into 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 $\scriptstyle W_p(\mathbb{D})$ .

Sufficient conditions

Theorem^[2]^[3] (Klee) — Let $d$ be any^{[note 1]} metric on a vector space $X$ such that the topology $��$ induced by $d$ on $X$ makes $(X, ��)$ into a topological vector space. If $(X, d)$ is a complete metric space then $(X, ��)$ is a complete-TVS.

Related properties

A linear almost continuous map into an F-space whose graph is closed is continuous.^[4]
A linear almost open map into an F-space whose graph is closed is necessarily an open map.^[4]
A linear continuous almost open map from an F-space is necessarily an open map.^[5]
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.^[4]

References

^ Not assume to be translation-invariant.

^ Dunford N., Schwartz J.T. (1958). Linear operators. Part I: general theory. Interscience publishers, inc., New York. p. 59
^ Schaefer & Wolff 1999, p. 35.
^ Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. 3 (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4.
^ ^a ^b ^c 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. Lecture Notes in Mathematics. 936. Berlin, Heidelberg, 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
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

[2] Not assume to be translation-invariant.

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

[FOOTNOTESchaeferWolff199935-3] Schaefer & Wolff 1999, p. 35.

[Klee_Inv_metrics-4] Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. 3 (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4.

[FOOTNOTEHusain197814-5] Husain 1978, p. 14.

[FOOTNOTEHusain197815-6] Husain 1978, p. 15.

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