Bounded set (topological vector space)

Notation: For any set A and scalar s, let sA := { sa : a ∈ 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:

1. 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]
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 B ⊆ sV;
4. for every neighborhood V of the origin there exists a real r > 0 such that sB ⊆ V 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 (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]
7. for every sequence (b_n)^∞
   _n=1 in B, the sequence (1/n b_n)^∞
   _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:

9. p(B) is bounded for all p ∈ 𝒫.[1]
10. there exists a sequence of non-0 scalars (s_i)^∞
    _i=1 such that for every sequence (b_i)^∞
    _i=1 in B, the sequence (s_ib_i)^∞
    _i=1 is bounded in X (according to any defining condition other than this one).[1]
11. 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:

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

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

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

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

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 ⊆ M, then S is bounded in M if and only if it is bounded in X.[1]

• 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.

• 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.

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 ⊂ N.

• Bornivorous set – A set that can absorb any bounded subset
• Bounded function
• Bounded operator – A linear operator that sends bounded subsets to bounded subsets
• Bounding point
• Compact space – Topological notions of all points being "close"
• Local boundedness
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Totally bounded space
• Topological vector space – Vector space with a notion of nearness

• 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.