Ultrabornological space

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In functional analysis, a topological vector space (TVS) [math]\displaystyle{ X }[/math] is called ultrabornological if every bounded linear operator from [math]\displaystyle{ X }[/math] into another TVS is necessarily continuous. A general version of the closed graph theorem holds for ultrabornological spaces. Ultrabornological spaces were introduced by Alexander Grothendieck (Grothendieck [1955, p. 17] "espace du type (β)").[1]

Definitions

Let [math]\displaystyle{ X }[/math] be a topological vector space (TVS).

Preliminaries

A disk is a convex and balanced set. A disk in a TVS [math]\displaystyle{ X }[/math] is called bornivorous[2] if it absorbs every bounded subset of [math]\displaystyle{ X. }[/math]

A linear map between two TVSs is called infrabounded[2] if it maps Banach disks to bounded disks.

A disk [math]\displaystyle{ D }[/math] in a TVS [math]\displaystyle{ X }[/math] is called infrabornivorous if it satisfies any of the following equivalent conditions:

  1. [math]\displaystyle{ D }[/math] absorbs every Banach disks in [math]\displaystyle{ X. }[/math]

while if [math]\displaystyle{ X }[/math] locally convex then we may add to this list:

  1. the gauge of [math]\displaystyle{ D }[/math] is an infrabounded map;[2]

while if [math]\displaystyle{ X }[/math] locally convex and Hausdorff then we may add to this list:

  1. [math]\displaystyle{ D }[/math] absorbs all compact disks;[2] that is, [math]\displaystyle{ D }[/math] is "compactivorious".

Ultrabornological space

A TVS [math]\displaystyle{ X }[/math] is ultrabornological if it satisfies any of the following equivalent conditions:

  1. every infrabornivorous disk in [math]\displaystyle{ X }[/math] is a neighborhood of the origin;[2]

while if [math]\displaystyle{ X }[/math] is a locally convex space then we may add to this list:

  1. every bounded linear operator from [math]\displaystyle{ X }[/math] into a complete metrizable TVS is necessarily continuous;
  2. every infrabornivorous disk is a neighborhood of 0;
  3. [math]\displaystyle{ X }[/math] be the inductive limit of the spaces [math]\displaystyle{ X_D }[/math] as D varies over all compact disks in [math]\displaystyle{ X }[/math];
  4. a seminorm on [math]\displaystyle{ X }[/math] that is bounded on each Banach disk is necessarily continuous;
  5. for every locally convex space [math]\displaystyle{ Y }[/math] and every linear map [math]\displaystyle{ u : X \to Y, }[/math] if [math]\displaystyle{ u }[/math] is bounded on each Banach disk then [math]\displaystyle{ u }[/math] is continuous;
  6. for every Banach space [math]\displaystyle{ Y }[/math] and every linear map [math]\displaystyle{ u : X \to Y, }[/math] if [math]\displaystyle{ u }[/math] is bounded on each Banach disk then [math]\displaystyle{ u }[/math] is continuous.

while if [math]\displaystyle{ X }[/math] is a Hausdorff locally convex space then we may add to this list:

  1. [math]\displaystyle{ X }[/math] is an inductive limit of Banach spaces;[2]

Properties

Every locally convex ultrabornological space is barrelled,[2] quasi-ultrabarrelled space, and a bornological space but there exist bornological spaces that are not ultrabornological.

  • Every ultrabornological space [math]\displaystyle{ X }[/math] is the inductive limit of a family of nuclear Fréchet spaces, spanning [math]\displaystyle{ X. }[/math]
  • Every ultrabornological space [math]\displaystyle{ X }[/math] is the inductive limit of a family of nuclear DF-spaces, spanning [math]\displaystyle{ X. }[/math]

Examples and sufficient conditions

The finite product of locally convex ultrabornological spaces is ultrabornological.[2] Inductive limits of ultrabornological spaces are ultrabornological.

Every Hausdorff sequentially complete bornological space is ultrabornological.[2] Thus every complete Hausdorff bornological space is ultrabornological. In particular, every Fréchet space is ultrabornological.[2]

The strong dual space of a complete Schwartz space is ultrabornological.

Every Hausdorff bornological space that is quasi-complete is ultrabornological.[citation needed]

Counter-examples

There exist ultrabarrelled spaces that are not ultrabornological. There exist ultrabornological spaces that are not ultrabarrelled.

See also

External links

References