Barrelled space


In functional analysis and related areas of mathematics, a barrelled space is a Hausdorff topological vector spaces 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 be a topological vector space.

Properties of barrels

Characterizations of barreled spaces

If is a topological vector space with continuous dual 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 showed 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 some point of .
  3. For any 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.
    • Recall that a linear map is called closed if its graph is a closed subset of.
  6. Every Hausdorff TVS topology on that has a neighborhood basis of 0 consisting of -closed set is course than.
and if is locally convex space then we may add to this list:

  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 principal holds.
  3. Every -bounded subset of the continuous dual space is equicontinuous ;
  4. carries the strong topology ;
  5. Every lower semicontinuous seminorm on is continuous;
  6. Every linear map into a locally convex space is almost continuous;
    • this means that for every neighborhood of 0 in, the closure of is a neighborhood of 0 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 ;
while if is a Hausdorff locally convex space then we may add to this list:

  1. Closed graph theorem: Every closed linear operator into a Banach space is continuous;
    • a closed linear operator is a linear operator whose graph is closed in.
  2. for all subsets 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;
while if is metrizable TVS then we may add to this list:

  1. For any complete metrizable TVS, every pointwise bounded sequence in is equicontinuous;
while if is a locally convex metrizable TVS then we may add to this list:

  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.

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 Lp space| is topologized as a subspace of, then it is not barrelled.
  3. Complete pseudometrizable TVSs.
  4. Montel spaces.
  5. Strong duals of Montel spaces.
  6. A locally convex quasi-barreled space that is also a ?-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 ultrabelled 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 sequentially complete quasibarrelled boundedly summing TVS.

Examples

;Counter examples

Properties of barreled spaces

Banach-Steinhaus Generalization

The importance of barrelled spaces is due mainly to the following results.
The Banach-Steinhaus theorem is a corollary of the above result. When the vector space consists of the complex numbers then the following generalization also holds.
Recall that a linear map is called closed if its graph is a closed subset of.

Other properties

History

Barrelled spaces were introduced by.