Barrelled space


In functional analysis and related areas of mathematics, a barrelled space is a topological vector space 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.
Barrelled spaces were introduced by.

Barrels

A convex and balanced subset of a real or complex vector space is called a and it is said to be,, or.
A ' or a ' in a topological vector space is a subset that is a closed absorbing disk; that is, a barrel is a convex, balanced, closed, and absorbing subset.
Every barrel must contain the origin. If and if is any subset of then is a convex, balanced, and absorbing set of if and only if this is all true of in for every -dimensional vector subspace thus if then the requirement that a barrel be a closed subset of is the only defining property that does not depend on -dimensional vector subspaces of
If is any TVS then every closed convex and balanced neighborhood of the origin is necessarily a barrel in . In fact, every locally convex topological vector space has a neighborhood basis at its origin consisting entirely of barrels. However, in general, there exist barrels that are not neighborhoods of the origin; "barrelled spaces" are exactly those TVSs in which every barrel is necessarily a neighborhood of the origin. Every finite dimensional topological vector space is a barrelled space so examples of barrels that are not neighborhoods of the origin can only be found in infinite dimensional spaces.

Examples of barrels and non-barrels

The closure of any convex, balanced, and absorbing subset is a barrel. This is because the closure of any convex subset has this same property.
A family of examples: Suppose that is equal to or equal to . Regardless of whether is a real or complex vector space, every barrel in is necessarily a neighborhood of the origin. Let be any function and for every angle let denote the closed line segment from the origin to the point Let Then is always an absorbing subset of but it is an absorbing subset of if and only if it is a neighborhood of the origin. Moreover, is a balanced subset of if and only if for every but is a balanced subset of if and only it is an open or closed ball centered at the origin. In particular, barrels in are exactly those closed balls centered at the origin with radius in If then is a closed subset that is absorbing in but not absorbing in and that is neither convex, balanced, nor a neighborhood of the origin in By an appropriate choice of the function it is also possible to have be a balanced and absorbing subset of that is neither closed nor convex. To have be a balanced, absorbing, and closed subset of that is convex nor a neighborhood of the origin, define on as follows: for let and then extend to by defining which guarantees that is balanced in

Properties of barrels


  • In any topological vector space every barrel in absorbs every compact convex subset of
  • In any locally convex Hausdorff TVS every barrel in absorbs every convex bounded complete subset of
  • If is locally convex then a subset of is -bounded if and only if there exists a barrel in such that
  • Let be a pairing and let be a locally convex topology on consistent with duality. Then a subset of is a barrel in if and only if is the polar of some -bounded subset of
  • Suppose is a vector subspace of finite codimension in a locally convex space and If is a barrel in then there exists a barrel in such that

Characterizations of barreled spaces

Denote by the space of continuous linear maps from into
If is a Hausdorff topological vector space with continuous dual space then the following are equivalent:

  1. is barrelled.
  2. ': Every barrel in is a neighborhood of the origin.
    • This definition is similar to a characterization of Baire TVSs proved by Saxon , who proved that a TVS 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 point of .
  3. For any Hausdorff TVS every pointwise bounded subset of is equicontinuous.
  4. For any F-space every pointwise bounded subset of is equicontinuous.
  5. Every closed linear operator from into a complete metrizable TVS is continuous.
  6. Every Hausdorff TVS topology on that has a neighborhood basis of the origin consisting of -closed set is coarser than
If is locally convex space then this list may be extended by appending:

  1. There exists a TVS not carrying the indiscrete topology such that every pointwise bounded subset of is equicontinuous.
  2. For any locally convex TVS every pointwise bounded subset of is equicontinuous.
    • 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 principle holds.
  3. Every -bounded subset of the continuous dual space is equicontinuous.
  4. carries the strong dual topology
  5. Every lower semicontinuous seminorm on is continuous.
  6. Every linear map into a locally convex space is almost continuous.
    • A linear map is called ' if for every neighborhood of the origin in the closure of is a neighborhood of the origin in
  7. Every surjective linear map from a locally convex space is almost open.
    • This means that for every neighborhood of 0 in the closure of is a neighborhood of 0 in
  8. If is a locally convex topology on such that has a neighborhood basis at the origin consisting of -closed sets, then is weaker than
If is a Hausdorff locally convex space then this list may be extended by appending:

  1. Closed graph theorem: Every closed linear operator into a Banach space is continuous.
  2. For every subset of the continuous dual space of the following properties are equivalent: is

    1. equicontinuous;
    2. relatively weakly compact;
    3. strongly bounded;
    4. weakly bounded.
  3. The 0-neighborhood bases in and the fundamental families of bounded sets in correspond to each other by polarity.
If is metrizable topological vector space then this list may be extended by appending:

  1. For any complete metrizable TVS every pointwise bounded in is equicontinuous.
If is a locally convex metrizable topological vector space then this list may be extended by appending:

  1. : The weak* topology on is sequentially complete.
  2. : Every weak* bounded subset of is -relatively countably compact.
  3. : Every countable weak* bounded subset of is equicontinuous.
  4. : is not the union of an increase sequence of nowhere dense disks.

Examples and sufficient conditions

Each of the following topological vector spaces is barreled:

  1. TVSs that are Baire space.
    • Consequently, 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 exist normed vector spaces that are barrelled. For example, if the -space is topologized as a subspace of then it is not barrelled.
  3. Complete pseudometrizable TVSs.
    • Consequently, every finite-dimensional TVS is barrelled.
  4. Montel spaces.
  5. Strong dual spaces of Montel spaces.
  6. A locally convex quasi-barrelled space that is also a σ-barrelled space.
  7. A sequentially complete quasibarrelled space.
  8. A quasi-complete Hausdorff locally convex infrabarrelled space.
    • A TVS is called quasi-complete if every closed and bounded subset is complete.
  9. A TVS with a dense barrelled vector subspace.
    • Thus the completion of a barreled space is barrelled.
  10. A Hausdorff locally convex TVS with a dense infrabarrelled vector subspace.
    • Thus the completion of an infrabarrelled Hausdorff locally convex space is barrelled.
  11. A vector subspace of a barrelled space that has countable codimensional.
    • In particular, a finite codimensional vector subspace of a barrelled space is barreled.
  12. A locally convex ultrabarelled TVS.
  13. A Hausdorff locally convex TVS such that every weakly bounded subset of its continuous dual space is equicontinuous.
  14. A locally convex TVS such that for every Banach space a closed linear map of into is necessarily continuous.
  15. A product of a family of barreled spaces.
  16. A locally convex direct sum and the inductive limit of a family of barrelled spaces.
  17. A quotient of a barrelled space.
  18. A Hausdorff sequentially complete quasibarrelled boundedly summing TVS.
  19. A locally convex Hausdorff reflexive space is barrelled.

Counterexamples


  • 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.
  • A closed subspace of a barreled space is not necessarily countably quasi-barreled.
  • There exists a dense vector subspace of the Fréchet barrelled space that is not barrelled.
  • There exist complete locally convex TVSs that are not barrelled.
  • The finest locally convex topology on an infinite-dimensional vector space is a Hausdorff barrelled space that is a meagre subset of itself.