ultrabornological space

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

A disk is a convex and balanced set.

A disk in a TVS $X$ is called bornivorous{{sfn|Narici|Beckenstein|2011|pp=441-457}} if it absorbs every bounded subset of $X.$

A linear map between two TVSs is called infrabounded{{sfn|Narici|Beckenstein|2011|pp=441-457}} 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;{{sfn|Narici|Beckenstein|2011|pp=441-457}}

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

3. $D$ absorbs all compact disks;{{sfn|Narici|Beckenstein|2011|pp=441-457}} that is, $D$ is "compactivorious".

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;{{sfn|Narici|Beckenstein|2011|pp=441-457}}

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 {{mvar|D}} 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 $u\; :\; X\; \backslash to\; Y,$ if $u$ is bounded on each Banach disk then $u$ is continuous;
7. for every Banach space $Y$ and every linear map $u\; :\; X\; \backslash to\; Y,$ 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;{{sfn|Narici|Beckenstein|2011|pp=441-457}}

Every locally convex ultrabornological space is barrelled,{{sfn|Narici|Beckenstein|2011|pp=441-457}} 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 $X.$

The finite product of locally convex ultrabornological spaces is ultrabornological.{{sfn|Narici|Beckenstein|2011|pp=441-457}} Inductive limits of ultrabornological spaces are ultrabornological.

Every Hausdorff sequentially complete bornological space is ultrabornological.{{sfn|Narici|Beckenstein|2011|pp=441-457}} Thus every complete Hausdorff bornological space is ultrabornological. In particular, every Fréchet space is ultrabornological.{{sfn|Narici|Beckenstein|2011|pp=441-457}}

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

Every Hausdorff bornological space that is quasi-complete is ultrabornological.{{citation needed|date=September 2020}}

;Counter-examples

There exist ultrabarrelled spaces that are not ultrabornological.

There exist ultrabornological spaces that are not ultrabarrelled.

• {{annotated link|Bounded linear operator}}
• {{annotated link|Bounded set (topological vector space)}}
• {{annotated link|Bornological space}}
• {{annotated link|Bornology}}
• {{annotated link|Locally convex topological vector space}}
• {{annotated link|Space of linear maps}}
• {{annotated link|Topological vector space}}
• {{annotated link|Vector bornology}}

• [http://www.numdam.org/article/AIF_1974__24_3_57_0.pdf Some characterizations of ultrabornological spaces]

{{reflist|group=note}}

{{reflist}}

• {{cite book

| last = Hogbe-Nlend

| first = Henri

| title = Bornologies and functional analysis

| publisher = North-Holland Publishing Co.

| location = Amsterdam

| year = 1977

| pages = xii+144

| isbn = 0-7204-0712-5

| mr = 0500064

}}

• {{Edwards Functional Analysis Theory and Applications}}
• {{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}}
• {{Grothendieck Topological Vector Spaces}}
• {{Khaleelulla Counterexamples in Topological Vector Spaces}}
• {{Kriegl Michor The Convenient Setting of Global Analysis}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}

{{Functional analysis}}

{{Boundedness and bornology}}

{{Topological vector spaces}}

Category:Topological vector spaces