Ultrabornological space

In functional analysis, a topological vector space (TVS) $X$ is called ultrabornological if every bounded linear operator from $X$ 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)").Template:Sfn

Definitions

Let $X$ be a topological vector space (TVS).

Preliminaries

A disk is a convex and balanced set. A disk in a TVS $X$ is called bornivorousTemplate:Sfn if it absorbs every bounded subset of $X.$

A linear map between two TVSs is called infraboundedTemplate:Sfn if it maps Banach disks to bounded disks.

A disk $D$ in a TVS $X$ is called infrabornivorous if it satisfies any of the following equivalent conditions:

1. $D$ absorbs every Banach disks in $X.$

while if $X$ locally convex then we may add to this list:

2. the gauge of $D$ is an infrabounded map;Template:Sfn

while if $X$ locally convex and Hausdorff then we may add to this list:

3. $D$ absorbs all compact disks;Template:Sfn that is, $D$ is "compactivorious".

Ultrabornological space

A TVS $X$ is ultrabornological if it satisfies any of the following equivalent conditions:

1. every infrabornivorous disk in $X$ is a neighborhood of the origin;Template:Sfn

while if $X$ is a locally convex space then we may add to this list:

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

while if $X$ is a Hausdorff locally convex space then we may add to this list:

8. $X$ is an inductive limit of Banach spaces;Template:Sfn

Properties

Every locally convex ultrabornological space is barrelled,Template:Sfn quasi-ultrabarrelled space, and a bornological space but there exist bornological spaces that are not ultrabornological.

• Every ultrabornological space $X$ is the inductive limit of a family of nuclear Fréchet spaces, spanning $X.$
• Every ultrabornological space $X$ is the inductive limit of a family of nuclear DF-spaces, spanning $X.$

Examples and sufficient conditions

The finite product of locally convex ultrabornological spaces is ultrabornological.Template:Sfn Inductive limits of ultrabornological spaces are ultrabornological.

Every Hausdorff sequentially complete bornological space is ultrabornological.Template:Sfn Thus every complete Hausdorff bornological space is ultrabornological. In particular, every Fréchet space is ultrabornological.Template:Sfn

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

Every Hausdorff bornological space that is quasi-complete is ultrabornological.Template:Citation needed

Counter-examples

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

See also

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External links

• Some characterizations of ultrabornological spaces

References

• Template:Cite book
• Template:Edwards Functional Analysis Theory and Applications
• Template:Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires
• Template:Grothendieck Topological Vector Spaces
• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Kriegl Michor The Convenient Setting of Global Analysis
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Wilansky Modern Methods in Topological Vector Spaces