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).

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.

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

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 topological vector space (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 is -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 Hausdorff 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]

Examples and 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. F-spaces, 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]
  19. A locally convex Hausdorff reflexive space is barrelled.

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 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] — 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 Br-complete TVS is necessarily continuous.[13]

History

Barrelled spaces were introduced by Bourbaki (1950).

See also

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

Bibliography
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  • Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). doi:10.5802/aif.16. MR 0042609.
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