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 Lp 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
is an F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.
Example 2
Let be the space of all complex valued Taylor series
on the unit disc such that
then (for 0 < p < 1) are F-spaces under the p-norm:
In fact, is a quasi-Banach algebra. Moreover, for any with the map is a bounded linear (multiplicative functional) on .
Sufficient conditions
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
- 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.
- 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.