Barrelled space

In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector spaces (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

Barrels

Let X be a topological vector space (TVS).

Definition: A convex and balanced subset of a real or complex vector space is called a disk and it is said to be disked, absolutely convex, or convex balanced.

Definition: A barrel or a barrelled set in a TVS is a subset that is a closed absorbing disk.

Note that the only topological requirement on a barrel is that it be a closed subset of the TVS; all other requirements (i.e. being a disk and being absorbing) are purely algebraic properties.

Properties of barrels

• In any TVS X, every barrel in X absorbs every compact convex subset of X.[1]
• In any locally convex Hausdorff TVS X, every barrel in X absorbs every convex bounded complete subset of X.[1]
• If X is locally convex then a subset H of X' is 𝜎(X', X)-bounded if and only if there exists a barrel B in X such that H ⊆ B°.[1]
• Let (X, Y, b) be a pairing and let 𝜏 be a locally convex topology on X consistent with duality. Then a subset B of X is a barrel in (X, 𝜏) if and only if B is the polar of some 𝜎(Y, X, b)-bounded subset of Y.[1]
• Suppose M is a vector subspace of finite codimension in a locally convex space X and B ⊆ M. If B is a barrel (resp. bornivorous barrel, bornivorous disk) in M then there exists a barrel (resp. bornivorous barrel, bornivorous disk) C in X such that B = C ∩ M.[2]

Characterizations of barreled spaces

Notation: Let L(X; Y) denote the space of continuous linear maps from X into Y.

If (X, 𝜏) is a topological vector space (TVS) with continuous dual X' then the following are equivalent:

1. X is barrelled;
2. (definition) Every barrel in X is a neighborhood of the origin;
   • This definition is similar to a characterization of Baire TVSs proved by Saxon [1974], who showed that a TVS Y with a topology that is not the indiscrete topology is a Baire space if and only if every absorbing balanced subset is a neighborhood of some point of Y (not necessarily the origin).[2]

If (X, 𝜏) is Hausdorff then we may add to this list:

3. For any TVS Y, every pointwise bounded subset of L(X; Y) is equicontinuous;[3]
4. For any F-space Y, every pointwise bounded subset of L(X; Y) is equicontinuous;[3]
   • An F-space is a complete metrizable TVS.
5. Every closed linear operator from X into a complete metrizable TVS is continuous.[4]
   • Recall that a linear map F : X → Y is called closed if its graph is a closed subset of X × Y.
6. Every Hausdorff TVS topology 𝜐 on X that has a neighborhood basis of 0 consisting of 𝜏-closed set is course than 𝜏.[5]

If (X, 𝜏) is locally convex space then we may add to this list:

7. There exists a TVS Y not carrying the indiscrete topology (so in particular, Y ≠ { 0 }) such that every pointwise bounded subset of L(X; Y) is equicontinuous;[2]
8. For any locally convex TVS Y, every pointwise bounded subset of L(X; Y) is equicontinuous;[2]
   • It follows from the above two characterizations that in the class of locally convex TVS, barrelled spaces are exactly those for which the uniform boundedness principal holds.
9. Every σ(X', X)-bounded subset of the continuous dual space X is equicontinuous (this provides a partial converse to the Banach-Steinhaus theorem);[2][6]
10. X carries the strong topology β(X, X');[2]
11. Every lower semicontinuous seminorm on X is continuous;[2]
12. Every linear map F : X → Y into a locally convex space Y is almost continuous;[2]
    • this means that for every neighborhood V of 0 in Y, the closure of F ^-1(V) is a neighborhood of 0 in X;
13. Every surjective linear map F : Y → X from a locally convex space Y is almost open;[2]
    • this means that for every neighborhood V of 0 in Y, the closure of F(V) is a neighborhood of 0 in X;
14. If ϖ is a locally convex topology on X such that (X, ϖ) has a neighborhood basis at the origin consisting of 𝜏-closed sets, then ϖ is weaker than 𝜏;[2]

If X is a Hausdorff locally convex space then we may add to this list:

15. Closed graph theorem: Every closed linear operator F : X → Y into a Banach space Y is continuous;[7]
    • a closed linear operator is a linear operator whose graph is closed in X × Y.
16. for all subsets A of the continuous dual space of X, the following properties are equivalent: A is [6]
    1. equicontinuous;
    2. relatively weakly compact;
    3. strongly bounded;
    4. weakly bounded;
17. the 0-neighborhood bases in X and the fundamental families of bounded sets in E_β' correspond to each other by polarity;[6]

If X is metrizable TVS then we may add to this list:

18. For any complete metrizable TVS Y, every pointwise bounded sequence in L(X; Y) is equicontinuous;[3]

If X is a locally convex metrizable TVS then we may add to this list:

19. (property S): the weak* topology on X' is sequentially complete;[8]
20. (property C): every weak* bounded subset of X' is 𝜎(X ', X)-relatively countably compact;[8]
21. (𝜎-barrelled): every countable weak* bounded subset of X' is equicontinuous;[8]
22. (Baire-like): X is not the union of an increase sequence of nowhere dense disks.[8]

Sufficient conditions

Each of the following topological vector spaces is barreled:

1. TVSs that are Baire space.
   • thus, also every topological vector space that is of the second category in itself is barrelled.
2. Fréchet spaces, Banach spaces, and Hilbert spaces.
   • However, there are normed vector spaces that are not barrelled. For instance, if L²([0, 1]) is topologized as a subspace of L¹([0, 1]), then it is not barrelled.
3. Complete pseudometrizable TVSs.[9]
4. Montel spaces.
5. Strong duals of Montel spaces (since they are Montel spaces).
6. A locally convex quasi-barreled space that is also a 𝜎-barrelled space.[10]
7. A sequentially complete quasibarrelled space.
8. A quasi-complete Hausdorff locally convex infrabarrelled space.[2]
   • A TVS is called quasi-complete if every closed and bounded subset is complete.
9. A TVS with a dense barrelled vector subspace.[2]
   • Thus the completion of a barreled space is barrelled.
10. A Hausdorff locally convex TVS with a dense infrabarrelled vector subspace.[2]
    • Thus the completion of an infrabarrelled Hausdorff locally convex space is barrelled.[2]
11. A vector subspace of a barrelled space that has countable codimensional.[2]
    • In particular, a finite codimensional vector subspace of a barrelled space is barreled.
12. A locally convex ultrabelled TVS.[11]
13. A Hausdorff locally convex TVS X such that every weakly bounded subset of its continuous dual space is equicontinuous.[12]
14. A locally convex TVS X such that for every Banach space B, a closed linear map of X into B is necessarily continuous.[13]
15. A product of a family of barreled spaces.[14]
16. A locally convex direct sum and the inductive limit of a family of barrelled spaces.[15]
17. A quotient of a barrelled space.[16][15]
18. A Hausdorff sequentially complete quasibarrelled boundedly summing TVS.[17]

Examples

Counter examples

• A barrelled space need not be Montel, complete, metrizable, unordered Baire-like, nor the inductive limit of Banach spaces.
• Not all normed spaces are barrelled. However, they are all infrabarrelled.[2]
• A closed subspace of a barreled space is not necessarily countably quasi-barreled (and thus not necessarily barrelled).[18]
• There exists a dense vector subspace of the Fréchet barrelled space ℝ^ℕ that is not barrelled.[2]
• There exist complete locally convex TVSs that are not barrelled.[2]
• The finest locally convex topology on a vector space is Hausdorff barrelled space that is a meagre subset of itself (and thus not a Baire space).[2]

Properties of barreled spaces

Banach-Steinhaus Generalization

The importance of barrelled spaces is due mainly to the following results.

Theorem[19] — Let X be a barrelled TVS and Y be a locally convex TVS. Let H be a subset of the space L(X; Y) of continuous linear maps from X into Y. The following are equivalent:

1. H is bounded for the topology of pointwise convergence;
2. H is bounded for the topology of bounded convergence;
3. H is equicontinuous.

The Banach-Steinhaus theorem is a corollary of the above result.[20] When the vector space Y consists of the complex numbers then the following generalization also holds.

Theorem[21] — If X is a barrelled TVS over the complex numbers and H is a subset of the continuous dual space of X, then the following are equivalent:

1. H is weakly bounded;
2. H is strongly bounded;
3. H is equicontinuous;
4. H is relatively compact in the weak dual topology.

Recall that a linear map F : X → Y is called closed if its graph is a closed subset of X × Y.

Closed Graph Theorem[22] — Every closed linear operator from a Hausdorff barrelled TVS into a complete metrizable TVS is continuous.

Other properties

• Every Hausdorff barrelled space is quasi-barrelled.[23]
• A linear map from a barrelled space into a locally convex space is almost continuous.
• A linear map from a locally convex space onto a barrelled space is almost open.
• A separately continuous bilinear map from a product of barrelled spaces into a locally convex space is hypocontinuous.[24]
• A linear map with a closed graph from a barreled TVS into a B_r-complete TVS is necessarily continuous.[13]

History

Barrelled spaces were introduced by Bourbaki (1950).

See also

• Countably barrelled space
• Infrabarreled space
• Quasibarrelled space
• Ultrabarrelled space
• Uniform boundedness principle#Generalisations

References

1. Narici 2011, pp. 225-273. sfn error: no target: CITEREFNarici2011 (help)
2. Narici 2011, pp. 371-423. sfn error: no target: CITEREFNarici2011 (help)
3. Adasch 1978, p. 39. sfn error: no target: CITEREFAdasch1978 (help)
4. Adasch 1978, p. 43. sfn error: no target: CITEREFAdasch1978 (help)
5. Adasch 1978, p. 32. sfn error: no target: CITEREFAdasch1978 (help)
6. Schaefer (1999) p. 127, 141, Trèves (1995) p. 350
7. Narici 2011, p. 477. sfn error: no target: CITEREFNarici2011 (help)
8. Narici 2011, p. 399. sfn error: no target: CITEREFNarici2011 (help)
9. Narici 2011, p. 383. sfn error: no target: CITEREFNarici2011 (help)
10. Khaleelulla 1982, pp. 28-63. sfn error: no target: CITEREFKhaleelulla1982 (help)
11. Narici 2011, pp. 418-419. sfn error: no target: CITEREFNarici2011 (help)
12. Trèves 2006, p. 350. sfn error: no target: CITEREFTrèves2006 (help)
13. Schaefer 1999, p. 166. sfn error: no target: CITEREFSchaefer1999 (help)
14. Schaefer 1999, p. 138. sfn error: no target: CITEREFSchaefer1999 (help)
15. Schaefer 1999, p. 61. sfn error: no target: CITEREFSchaefer1999 (help)
16. Trèves 2006, p. 346. sfn error: no target: CITEREFTrèves2006 (help)
17. Adasch 1978, p. 77. sfn error: no target: CITEREFAdasch1978 (help)
18. Schaefer 1999, pp. 103-110. sfn error: no target: CITEREFSchaefer1999 (help)
19. Trèves 2006, p. 347. sfn error: no target: CITEREFTrèves2006 (help)
20. Trèves 2006, p. 348. sfn error: no target: CITEREFTrèves2006 (help)
21. Trèves 2006, p. 349. sfn error: no target: CITEREFTrèves2006 (help)
22. Adasch 1978, p. 41. sfn error: no target: CITEREFAdasch1978 (help)
23. Adasch 1978, pp. 70-73. sfn error: no target: CITEREFAdasch1978 (help)
24. Trèves 2006, p. 424. sfn error: no target: CITEREFTrèves2006 (help)

* Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. {3834. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.CS1 maint: ref=harv (link)

• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
• Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). MR 0042609.
• Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5 [Sur certains espaces vectoriels topologiques]. Annales de l'Institut Fourier. Elements of mathematics (in French). 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.CS1 maint: ref=harv (link)
• Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.CS1 maint: ref=harv (link)
• Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.CS1 maint: ref=harv (link)
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
• Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 65–75.
• 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.CS1 maint: ref=harv (link)
• Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.CS1 maint: ref=harv (link)
• Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)