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Ultrabornological space

## Summary

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:

1. 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:

1. ${\displaystyle D}$  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:

1. every bounded linear operator from ${\displaystyle X}$  into a complete metrizable TVS is necessarily continuous;
2. every infrabornivorous disk is a neighborhood of 0;
3. ${\displaystyle X}$  be the inductive limit of the spaces ${\displaystyle X_{D}}$  as D varies over all compact disks in ${\displaystyle X}$ ;
4. a seminorm on ${\displaystyle X}$  that is bounded on each Banach disk is necessarily continuous;
5. 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;
6. 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:

1. ${\displaystyle X}$  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.