Tubular space

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Tubular spaces are special locally convex vector spaces in which the Banach-Steinhaus theorem applies. These spaces can be characterized by their null environment bases.

motivation

A ton is a subset T of a locally convex K-vector space E with the following three properties

  • T is absolutely convex , i.e. H. for and with applies .
  • T is closed in the topology on E.
  • T is absorbent , i.e. H. for each there is a with .

It is easy to show that every locally convex space has a zero neighborhood basis of tons. Conversely, if every barrel is a zero environment, the space is called barreled . This name goes back to Bourbaki (French tonnelé, English barreled).

Examples

  • Every Fréchet room (especially every Banach room ) is barreled. If T is a ton in the Fréchet space E , then T is absorbent. Because T is closed, it follows from Baire's theorem that an nT and thus T has an interior point . From the absolute convexity of T it follows easily that T is a null neighborhood.
  • If E is a Fréchet space not equal to {0}, the product topology is an example of a barreled space that is not a Fréchet space.

Inheritance properties

Quotient spaces after closed sub-spaces, product spaces and inductive limits of barreled rooms are barreled again.

The tonnage is generally not passed on to closed subspaces or projective limits .

The Banach Stone House theorem

A set B in a locally convex space E is known to be called bounded if it is absorbed by every null neighborhood, i.e. H. for every zero neighborhood U of E there is a with . A family of continuous linear operators between locally convex vector spaces E and F is called uniformly continuous if for every null neighborhood V in F there is a null neighborhood U in E such that for all . The following sentence identifies the barreled rooms as those in which the Banach-Steinhaus Theorem applies:

For a locally convex space E are equivalent:

  • E is a barreled room.
  • If F is another locally convex space and a family of continuous linear operators that is pointwise bounded (i.e. for each is bounded), then is uniformly continuous.

Relationship to reflexivity

If E is a locally convex vector space, then every bounded set B in E defines a semi-norm on the dual space by setting. Provided with the set of semi-norms , where B runs through the bounded sets of E, becomes a locally convex vector space, which is then denoted by. This generalizes the dual space formation in normalized spaces . As there one has a natural embedding , and as usual one identifies E with J (E), so that E can be understood as a subspace of the bidual . If J is surjective , then E is called semi-reflective . Then E agree as sets, but in general the locally convex topologies on E and the bidual are different. If the topologies also match, E is called reflexive . The tonnage is precisely the quality that a semi-reflective space lacks for reflexivity, because the following applies:

For a locally convex space E are equivalent:

  • E is reflexive.
  • E is semi-reflective and barreled.

Quasitonné rooms

A set T in a locally convex space is called bornivor if it absorbs every bounded set, i.e. H. if for every bounded set B there is one with . (For the origin of the word, compare carnivorous or herbivorous .)

A space is called quasitonneliert if every bornivore barrel is a null environment. Obviously it is a weakening of the tonnage. Bornological spaces are quasitonneliert. Sequentially complete, quasi-barreled spaces are already barreled, as follows from Banach-Mackey's theorem.

The quasitonneliness is sufficient with the above characterization of the reflexivity, because for a locally convex space E the following are equivalent:

  • E is reflexive
  • E is semi-reflective and barreled.
  • E is semi-reflective and quasitonneliert.

swell

  • Klaus Floret, Joseph Wloka : Introduction to the theory of locally convex spaces (= Lecture Notes in Mathematics. Vol. 56, ISSN  0075-8434 ). Springer, Berlin et al. 1968, doi : 10.1007 / BFb0098549 .
  • Reinhold Meise, Dietmar Vogt: Introduction to functional analysis (= Vieweg study 62 advanced course in mathematics ). Vieweg, Braunschweig et al. 1992, ISBN 3-528-07262-8 .