Bounded set (topological vector space)

In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. A set that is not bounded is called unbounded.

Bounded sets are a natural way to define a locally convex polar topologies on the vector spaces in a dual pair, as the polar of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.

Definition

Notation: For any set

A

and scalar

s

, let

sA := {sa : a \in A

}.

Definition: Given a topological vector space (TVS) $(X, τ)$ over a field $𝕂$ , a subset $B$ of $X$ is called von Neumann bounded or just bounded in $X$ if any of the following equivalent conditions is satisfied:

for every neighborhood V of the origin there exists a real r > 0 such that B ⊆ sV for all scalars s satisfying |s| ≥ r;[1]
- This was the definition introduced by John von Neumann in 1935.[1]
$B$ is absorbed by every neighborhood of the origin;[2]
for every neighborhood $V$ of the origin there exists a scalar $s$ such that $B \subseteq sV$ ;
for every neighborhood $V$ of the origin there exists a real $r > 0$ such that $sB \subseteq V$ for all scalars $s$ satisfying $| s | \leq r$ ;[1]
Any of the above 4 conditions but with the word "neighborhood" replaced by any of the following: "balanced neighborhood," "open balanced neighborhood," "closed balanced neighborhood," "open neighborhood," "closed neighborhood";
- e.g. Condition 2 may become: $B$ is bounded if and only if $B$ is absorbed by every balanced neighborhood of the origin.[1]
for every sequence of scalars (s_i)^∞
_i=1> that converges to 0 and every sequence (b_i)^∞
_i=1 in B, the sequence (s_i b_i)^∞
_i=1 converges to 0 in X;[1]
- This was the definition of "bounded" that Andrey Kolmogorov used in 1934, which is the same as the definition introduced by Stanisław Mazur and Władysław Orlicz in 1933 for metrizable TVS. Kolmogorov used this definition to prove that a TVS is seminormable if and only if it has a bounded convex neighborhood of 0.[1]
for every sequence $(b n) \infty n =1$ in $B$ , the sequence $(1 / n b n) \infty n =1 \to 0$ in $X$ ;[3]
every countable subset of $B$ is bounded (according to any defining condition other than this one).[1]

while if $X$ is a locally convex space whose topology is defined by a family $𝒫$ of continuous seminorms, then we may add to this list:

$p (B)$ is bounded for all $p \in 𝒫$ .[1]
there exists a sequence of non-0 scalars $(s i) \infty i =1$ such that for every sequence $(b i) \infty i =1$ in $B$ , the sequence $(s i b i) \infty i =1$ is bounded in $X$ (according to any defining condition other than this one).[1]
for all $p \in 𝒫$ , $B$ is bounded (according to any defining condition other than this one) in the semi normed space $(X, p)$ .

while if $X$ is a seminormed space with seminorm $p$ (note that every normed space is a seminormed space and every norm is a seminorm), then we may add to this list:

There exists a real $r > 0$ such that $p (b) \leq r$ for all $b \in B$ .[1]

while if $B$ is a vector subspace of the TVS $X$ then we may add to this list:

$B$ is contained in the closure of ${ 0$ }.[1]

Definition: A subset that is not bounded is called unbounded.

Bornology and fundamental systems of bounded sets

The collection of all bounded sets on a topological vector space $X$ is called the von Neumann bornology or the (canonical) bornology of $X$ .

A base or fundamental system of bounded sets of $X$ is a set $ℬ$ of bounded subsets of $X$ such that every bounded subset of $X$ is a subset of some $B \in ℬ$ .[1] The set of all bounded subsets of $X$ trivially forms a fundamental system of bounded sets of $X$ .

Examples

In any locally convex TVS, the set of closed and bounded disks are a base of bounded set.[1]

Stability properties

Let $X$ be any topological vector space (TVS) (not necessarily Hausdorff or locally convex).

In any TVS, finite unions, finite sums, scalar multiples, subsets, closures, interiors, and balanced hulls of bounded sets are again bounded.[1]
In any locally convex TVS, the convex hull of a bounded set is again bounded. This may fail to be true if the space is not locally convex.[1]
The image of a bounded set under a continuous linear map is a bounded subset of the codomain.[1]
A subset of an arbitrary product of TVSs is bounded if and only if all of its projections are bounded.
If $M$ is a vector subspace of a TVS $X$ and if $S \subseteq M$ , then $S$ is bounded in $M$ if and only if it is bounded in $X$ .[1]

Examples and sufficient conditions

In any topological vector space (TVS), finite sets are bounded.[1]
Every totally bounded subset of a TVS is bounded.[1]
Every relatively compact set in a topological vector space is bounded. If the space is equipped with the weak topology the converse is also true.
The set of points of a Cauchy sequence is bounded, the set of points of a Cauchy net need not be bounded.
In any TVS, every subset of the closure of ${ 0$ } is bounded.

Non-examples

In any TVS, any vector subspace that is not a contained in the closure of ${ 0$ } is unbounded (i.e. not bounded).
There exists a Fréchet space $X$ having a bounded subset $B$ and also a dense vector subspace $M$ such that $B$ is not contained in the closure (in $X$ ) of any bounded subset of $M$ .[4]

Properties

Finite unions, finite sums, closures, interiors, and balanced hulls of bounded sets are bounded.
The image of a bounded set under a continuous linear map is bounded.
In a locally convex space, the convex envelope of a bounded set is bounded.
- Without local convexity this is false, as the $L p$ spaces for $0 < p < 1$ have no nontrivial open convex subsets.
A locally convex space has a bounded neighborhood of zero if and only if its topology can be defined by a single seminorm.
The polar of a bounded set is an absolutely convex and absorbing set.

Mackey's countability condition ([1]) — Suppose that $X$ is a metrizable locally convex TVS and that $(B i) \infty i =1$ is a countable sequence of bounded subsets of $X$ . Then there exists a bounded subset $B$ of $X$ and a sequence $(r i) \infty i =1$ of positive real numbers such that $B i \subseteq r i B$ for all $i$ .

Generalization

The definition of bounded sets can be generalized to topological modules. A subset $A$ of a topological module $M$ over a topological ring $R$ is bounded if for any neighborhood $N$ of $0 M$ there exists a neighborhood $w$ of $0 R$ such that $w A \subset N$ .

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References

Narici 2011, pp. 156-175.
Schaefer 1970, p. 25.
Wilansky 2013, p. 47.
Wilansky 2013, p. 57.

Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. {3834. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.CS1 maint: ref=harv (link)
Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5 [Sur certains espaces vectoriels topologiques]. Annales de l'Institut Fourier. Elements of mathematics (in French). 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.CS1 maint: ref=harv (link)
Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.CS1 maint: ref=harv (link)
Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.CS1 maint: ref=harv (link)
Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.CS1 maint: ref=harv (link)
Narici, Lawrence (2011). Topological vector spaces. Boca Raton, FL: CRC Press. ISBN 1-58488-866-0. OCLC 144216834.
Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 44–46.
H.H. Schaefer (1970). Topological Vector Spaces. GTM. 3. Springer-Verlag. pp. 25–26. ISBN 0-387-05380-8.
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.CS1 maint: ref=harv (link)

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[FOOTNOTENarici2011156-175-1] Narici 2011, pp. 156-175.

[2] Schaefer 1970, p. 25.

[FOOTNOTEWilansky201347-3] Wilansky 2013, p. 47.

[FOOTNOTEWilansky201357-4] Wilansky 2013, p. 57.

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