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

1. Scalar multiplication in V is continuous with respect to d and the standard metric on ℝ or ℂ.
2. Addition in V is continuous with respect to d.
3. The metric is translation-invariant; i.e., d(x + a, y + a) = d(x, y) for all x, y and a in V
4. 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 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

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

Example 2

Let ${\displaystyle \scriptstyle W_{p}(\mathbb {D} )}$ be the space of all complex valued Taylor series

${\displaystyle f(z)=\sum _{n\geq 0}a_{n}z^{n}}$

on the unit disc ${\displaystyle \scriptstyle \mathbb {D} }$ such that

${\displaystyle \sum _{n}|a_{n}|^{p}<\infty }$

then (for 0 < p < 1) ${\displaystyle \scriptstyle W_{p}(\mathbb {D} )}$ are F-spaces under the p-norm:

${\displaystyle \|f\|_{p}=\sum _{n}|a_{n}|^{p}\qquad (0<p<1)}$

In fact, ${\displaystyle \scriptstyle W_{p}}$ is a quasi-Banach algebra. Moreover, for any ${\displaystyle \scriptstyle \zeta }$ with ${\displaystyle \scriptstyle |\zeta |\;\leq \;1}$ the map ${\displaystyle \scriptstyle f\,\mapsto \,f(\zeta )}$ is a bounded linear (multiplicative functional) on ${\displaystyle \scriptstyle W_{p}(\mathbb {D} )}$.

Sufficient conditions

Theorem[2][3] (Klee) — Let d be any[note 1] metric on a vector space X such that the topology 𝜏 induced by d on X makes (X, 𝜏) into a topological vector space. If (X, d) is a complete metric space then (X, 𝜏) 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 – Mathematical generalization to infinite dimension of the notion of Euclidean space
• K-space (functional analysis)
• Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
• Barreled space
• Countably quasi-barrelled space
• DF-space
• LB-space
• LF-space
• Nuclear space
• Projective tensor product

References

1. 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
2. Schaefer & Wolff 1999, p. 35.
3. 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.
4. Husain 1978, p. 14.
5. 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.