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 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
\( \scriptstyle L^\frac{1}{2}[0,\, 1] \)is an F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.
Example 2
Let W p ( D ) {\displaystyle \scriptstyle W_{p}(\mathbb {D} )} \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) W p ( D ) {\displaystyle \scriptstyle W_{p}(\mathbb {D} )} \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 \( \tau \) induced by d on X makes (X, \( \tau \)) into a topological vector space. If (X, d) is a complete metric space then (X, \( \tau \) ) 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]
See also
Banach space – Normed vector space that is complete
Complete metric space – A set with a notion of distance where every sequence of points that get progressively closer to each other will converge
Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
Fréchet space – A locally convex topological vector space that is also a complete metric space
Hilbert space – Inner product space that is metrically complete; a Banach space whose norm induces an inner product (The norm satisfies the parallelogram identity)
K-space (functional analysis)
Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
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): 484–487. doi:10.1090/s0002-9939-1952-0047250-4.
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. (July 1, 1982). Written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. 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
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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