Ultrabornological space

In functional analysis, a topological vector space (TVS) ${\displaystyle X}$ is called ultrabornological if every

from ${\displaystyle X}$ into another TVS is necessarily 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

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. ${\displaystyle D}$
   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

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

]

Counter-examples

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

See also

• 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

External links

• Some characterizations of ultrabornological spaces

References

• Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co. pp. xii+144.
  MR 0500064
  .
• Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications.
  OCLC 30593138
  .
• OCLC 1315788
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• OCLC 886098
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• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces.
  OCLC 8588370
  .
• OCLC 37141279
  .
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
  OCLC 144216834
  .
• OCLC 840278135
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• OCLC 849801114
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