Ultrabornological space

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.

Definitions

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

Preliminaries

Definition: A disk is a convex and balanced set.

Definition:[1] A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks.

Definition:[1] 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;[1]

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

D absorbs all compact disks.[1]
- 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;[1]

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 $X D$ 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 $u : X \to Y$ , if $u$ is bounded on each Banach disk then $u$ is continuous;
for every Banach space $Y$ and every linear map $u : X \to Y$ , 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:

$X$ is an inductive limit of Banach spaces;[1]

Properties

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 bornological space that is quasi-complete is ultrabornological.
- In particular, every Fréchet space is ultrabornological.
Every metrizable TVS is ultrabornological.
The finite product of ultrabornological spaces is ultrabornological.
Inductive limits of ultrabornological spaces are ultrabornological.

Counter-exmples

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

See also

Bornological space
Bounded linear operator
Bounded set (topological vector space)
Bornological space
Bornology
Locally convex topological vector space – Type of topological vector space
Space of linear maps
Topological vector space – Vector space with a notion of continuity
Vector bornology

External links

Some characterizations of ultrabornological spaces

References

Narici 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.
Khaleelulla, S. M. (July 1, 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.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
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.CS1 maint: ref=harv (link)
Khaleelulla, S.M. (1982). Counterexamples in Topological Vector Spaces. GTM. 936. Berlin Heidelberg: Springer-Verlag. pp. 29–33, 49, 104. ISBN 9783540115656.
Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. ISBN 9780821807804.

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

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 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

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