In functional analysis, a topological vector space (TVS) ${\displaystyle X}$ is called ultrabornological if every bounded linear operator from ${\displaystyle X}$ 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)").^[1]

Definitions

Let ${\displaystyle X}$ be a topological vector space (TVS).

Preliminaries

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

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

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

1. ${\displaystyle D}$ absorbs every Banach disks in ${\displaystyle X.}$

while if ${\displaystyle X}$ locally convex then we may add to this list:

2. the gauge of ${\displaystyle D}$ is an infrabounded map;^[2]

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

3. ${\displaystyle D}$ absorbs all compact disks;^[2] that is, ${\displaystyle D}$ is "compactivorious".

Ultrabornological space

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

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

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

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

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

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

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.