• Images
• Videos
• Documents
• Books
• Articles

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 ( a x , 0 ) = | a | d ( x , 0 ) . {\displaystyle d(ax,0)=|a|d(x,0).} ¹

The Lp spaces can be made into F-spaces for all p ≥ 0 {\displaystyle p\geq 0} and for p ≥ 1 {\displaystyle p\geq 1} they can be made into locally convex and thus Fréchet spaces and even Banach spaces.

Example 1

L 1 2 [ 0 , 1 ] {\displaystyle 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 W_{p}(\mathbb {D} )} be the space of all complex valued Taylor series f ( z ) = ∑ n ≥ 0 a n z n {\displaystyle f(z)=\sum _{n\geq 0}a_{n}z^{n}} on the unit disc D {\displaystyle \mathbb {D} } such that ∑ n | a n | p < ∞ {\displaystyle \sum _{n}\left|a_{n}\right|^{p}<\infty } then for 0 < p < 1 , {\displaystyle 0<p<1,} W p ( D ) {\displaystyle W_{p}(\mathbb {D} )} are F-spaces under the p-norm: ‖ f ‖ p = ∑ n | a n | p ( 0 < p < 1 ) . {\displaystyle \|f\|_{p}=\sum _{n}\left|a_{n}\right|^{p}\qquad (0<p<1).}

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

Sufficient conditions

Theorem²³ (Klee (1952))—Let d {\displaystyle d} be any⁴ metric on a vector space X {\displaystyle X} such that the topology τ {\displaystyle \tau } induced by d {\displaystyle d} on X {\displaystyle X} makes ( X , τ ) {\displaystyle (X,\tau )} into a topological vector space. If ( X , d ) {\displaystyle (X,d)} is a complete metric space then ( X , τ ) {\displaystyle (X,\tau )} is a complete topological vector space.

Related properties

The open mapping theorem implies that if τ  and  τ 2 {\displaystyle \tau {\text{ and }}\tau _{2}} are topologies on X {\displaystyle X} that make both ( X , τ ) {\displaystyle (X,\tau )} and ( X , τ 2 ) {\displaystyle \left(X,\tau _{2}\right)} into complete metrizable topological vector spaces (for example, Banach or Fréchet spaces) and if one topology is finer or coarser than the other then they must be equal (that is, if τ ⊆ τ 2  or  τ 2 ⊆ τ  then  τ = τ 2 {\displaystyle \tau \subseteq \tau _{2}{\text{ or }}\tau _{2}\subseteq \tau {\text{ then }}\tau =\tau _{2}} ).⁵

• A linear almost continuous map into an F-space whose graph is closed is continuous.⁶
• A linear almost open map into an F-space whose graph is closed is necessarily an open map.⁷
• A linear continuous almost open map from an F-space is necessarily an open map.⁸
• 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.⁹

See also

• Banach space – Normed vector space that is complete
• Barreled space – Type of topological vector spacePages displaying short descriptions of redirect targets
• Countably quasi-barrelled space
• Complete metric space – Metric geometry
• Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
• DF-space – class of special local-convex spacePages displaying wikidata descriptions as a fallback
• Fréchet space – Locally convex topological vector space that is also a complete metric space
• Hilbert space – Type of topological vector space
• K-space (functional analysis)
• LB-space
• LF-space – Topological vector space
• Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
• Nuclear space – A generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
• Projective tensor product – tensor product defined on two topological vector spacesPages displaying wikidata descriptions as a fallback

Notes

Sources

• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Rudin, Walter (1966). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

References

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. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

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. https://www.ams.org/journals/proc/1952-003-03/S0002-9939-1952-0047250-4/S0002-9939-1952-0047250-4.pdf ↩

4. Not assume to be translation-invariant. ↩

5. Trèves 2006, pp. 166–173. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

6. Husain & Khaleelulla 1978, p. 14. - Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665. https://search.worldcat.org/oclc/4493665 ↩

7. Husain & Khaleelulla 1978, p. 14. - Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665. https://search.worldcat.org/oclc/4493665 ↩

8. Husain & Khaleelulla 1978, p. 15. - Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665. https://search.worldcat.org/oclc/4493665 ↩

9. Husain & Khaleelulla 1978, p. 14. - Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665. https://search.worldcat.org/oclc/4493665 ↩