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:

  1. For any TVS Y, every pointwise bounded subset of L(X; Y) is equicontinuous;[3]
  2. For any F-space Y, every pointwise bounded subset of L(X; Y) is equicontinuous;[3]
    • An F-space is a complete metrizable TVS.
  3. 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.
  4. 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:

  1. 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]
  2. 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.
  3. 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]
  4. X carries the strong topology β(X, X');[2]
  5. Every lower semicontinuous seminorm on X is continuous;[2]
  6. 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;
  7. 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;
  8. 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:

  1. 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.
  2. 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;
  3. the 0-neighborhood bases in X and the fundamental families of bounded sets in Eβ' correspond to each other by polarity;[6]

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

  1. 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:

  1. (property S): the weak* topology on X' is sequentially complete;[8]
  2. (property C): every weak* bounded subset of X' is 𝜎(X ', X)-relatively countably compact;[8]
  3. (𝜎-barrelled): every countable weak* bounded subset of X' is equicontinuous;[8]
  4. (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. FrĂŠchet 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 FrĂŠchet barrelled space ℝℕ 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] â€” 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] â€” 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] â€” 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 Br-complete TVS is necessarily continuous.[13]

History

Barrelled spaces were introduced by Bourbaki (1950).

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gollark: It does not seem like a very good or useful description of anything.
gollark: Maybe they're just bad at describing universal constants.
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gollark: I may be reading off random browser tabs.

See also

References
  1. Narici 2011, pp. 225-273.
  2. Narici 2011, pp. 371-423.
  3. Adasch 1978, p. 39.
  4. Adasch 1978, p. 43.
  5. Adasch 1978, p. 32.
  6. Schaefer (1999) p. 127, 141, Trèves (1995) p. 350
  7. Narici 2011, p. 477.
  8. Narici 2011, p. 399.
  9. Narici 2011, p. 383.
  10. Khaleelulla 1982, pp. 28-63.
  11. Narici 2011, pp. 418-419.
  12. Trèves 2006, p. 350.
  13. Schaefer 1999, p. 166.
  14. Schaefer 1999, p. 138.
  15. Schaefer 1999, p. 61.
  16. Trèves 2006, p. 346.
  17. Adasch 1978, p. 77.
  18. Schaefer 1999, pp. 103-110.
  19. Trèves 2006, p. 347.
  20. Trèves 2006, p. 348.
  21. Trèves 2006, p. 349.
  22. Adasch 1978, p. 41.
  23. Adasch 1978, pp. 70-73.
  24. Trèves 2006, p. 424.
* 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)
  • KĂśthe, 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)
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