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In functional analysis, a topological vector space (TVS) X is called ultrabornological if every bounded linear operator from 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)").

Definitions

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

Definition: A disk is a convex and balanced set.

Definition: A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks.

Definition: A disk in a TVS X is called bornivorous if it absorbs every bounded subset of X.

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

D absorbs every Banach disks in X.

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

the guage of D is an infrabounded map;

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

D absorbs all compact disks.
i.e. D is "compactivorious".

Ultrabornological space

A TVS X is ultrabornological if it satisfies any of the following equivalent conditions:

every infrabornivorous disk in X is a neighborhood of the origin;

while if X is a locally convex space then we may add to this list:

every bounded linear operator from X into a complete metrizable TVS is necessarily continuous;
every infrabornivorous disk is a neighborhood of 0;
X be the inductive limit of the spaces XD as D varies over all compact disks in X;
a seminorm on X that is bounded on each Banach disk is necessarily continuous;
for every locally convex space Y and every linear map u : X → Y, if u is bounded on each Banach disk then u is continuous;
for every Banach space Y and every linear map u : X → 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:

X is an inductive limit of Banach spaces;

Properties

Every locally convex ultrabornological space is barrelled.
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.
Every ultrabornological space is a quasi-ultrabarrelled space.
Every locally convex ultrabornological space is a bornological space but there exist bornological spaces that are not ultrabornological.

Examples and sufficient conditions

Every Hausdorff sequentially complete bornological TVS is ultrabornological.
Thus every compete Hausdorff bornological space is ultrabornological.
In particular, every Fréchet space is ultrabornological.
The finite product of locally convex ultrabornological spaces is ultrabornological.
Inductive limits of ultrabornological spaces are ultrabornological.
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.

Bornological space – A topological vector space where any bounded linear operator into another space is always continuous
Bounded linear operator
Bounded set (topological vector space)
Bornological space – A topological vector space where any bounded linear operator into another space is always continuous
Bornology
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

References

Narici & Beckenstein 2011, p. 441.

Narici & Beckenstein 2011, pp. 441-457.

Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co. pp. xii+144. ISBN 0-7204-0712-5. MR 0500064.
Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). Providence: American Mathematical Society. 16. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788.
Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis (PDF). Mathematical Surveys and Monographs. 53. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-0780-4. OCLC 37141279.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

Functional analysis (topics – glossary)

Boundedness and Bornology
Basic concepts

Barrelled space Bounded set Bornological space (Vector) Bornology

Operators

(Un)Bounded operator

Subsets

Barrelled set Bornivorous set Saturated family

Operators

(Countably) Barrelled space (Countably) Quasi-barrelled space Ultrabarrelled space Ultrabornological space

Topological vector spaces (TVSs)
Basic concepts

Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space

Main results

Closed graph theorem F. Riesz's theorem Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) (Bounded inverse) Uniform boundedness (Banach–Steinhaus)

Maps

Almost open Bilinear (form operator) and Sesquilinear forms Closed Compact operator Continuous and Discontinuous Linear maps Densely defined Homomorphism Functionals Norm Operator Seminorm Sublinear Transpose

Types of sets

Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Radial Radially convex/Star-shaped Symmetric

Set operations

Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior

Types of TVSs

Asplund B-complete/Ptak Banach (Countably) Barrelled (Ultra-) Bornological Brauner Complete (DF)-space Distinguished F-space Fréchet (tame Fréchet) Grothendieck Hilbert Infrabarreled Interpolation space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property