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

Preliminaries edit

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 edit

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]

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 ${\displaystyle X.}$ 

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.

• Bounded linear operator – Linear transformation between topological vector spacesPages displaying short descriptions of redirect targets
• Bounded set (topological vector space) – Generalization of boundedness
• Bornological space – Space where bounded operators are continuous
• Bornology – Mathematical generalization of boundedness
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Space of linear maps
• Topological vector space – Vector space with a notion of nearness
• Vector bornology

• Some characterizations of ultrabornological spaces

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