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 : aA}.

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:

1. for every neighborhood V of the origin there exists a real r > 0 such that BsV for all scalars s satisfying |s| ≥ r;[1]
2. B is absorbed by every neighborhood of the origin;[2]
3. for every neighborhood V of the origin there exists a scalar s such that BsV;
4. for every neighborhood V of the origin there exists a real r > 0 such that sBV for all scalars s satisfying |s| ≤ r;[1]
5. 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]
6. for every sequence of scalars (si)
i=1
> that converges to 0 and every sequence (bi)
i=1
in B, the sequence (si bi)
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]
7. for every sequence (bn)
n=1
in B, the sequence (1/n bn)
n=1
→ 0
in X;[3]
8. 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:

1. p(B) is bounded for all p ∈ 𝒫.[1]
2. there exists a sequence of non-0 scalars (si)
i=1
such that for every sequence (bi)
i=1
in B, the sequence (sibi)
i=1
is bounded in X (according to any defining condition other than this one).[1]
3. for all p ∈ 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:

1. There exists a real r > 0 such that p(b) ≤ r for all bB.[1]

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

1. 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 of bounded subsets of X such that every bounded subset of X is a subset of some B ∈ 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 SM, then S is bounded in M if and only if it is bounded in X.[1]

Examples and sufficient conditions

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 Lp 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 (Bi)
i=1
is a countable sequence of bounded subsets of X. Then there exists a bounded subset B of X and a sequence (ri)
i=1
of positive real numbers such that Biri 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 0M there exists a neighborhood w of 0R such that w A ⊂ N.

References

1. Narici & Beckenstein 2011, pp. 156-175.
2. ^ Schaefer 1970, p. 25.
3. ^ Wilansky 2013, p. 47.
4. ^ Wilansky 2013, p. 57.

Bibliography

• Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
• Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
• Bourbaki, Nicolas (1987) [1981]. Sur certains espaces vectoriels topologiques [Topological Vector Spaces: Chapters 1–5]. Annales de l'Institut Fourier. Éléments de mathématique. 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.
• 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.
• Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
• 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.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• 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.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 44–46.
• Schaefer, H.H. (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.