Bounded set (topological vector space)

Given a topological vector space (TVS) (X,τ) over a field F, a subset B of X is called bounded 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] where sV := { sv : v ∈ V}.
   • This was the definition introduced by von Neumann in 1935.[1]
2. 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]
3. for every neighborhood V of the origin there exists a scalar s such that B ⊆ sV.
4. B is absorbed by every neighborhood of the origin,[2]
   • i.e., that for all neighborhoods N, there exists t such that

   ${\displaystyle |\alpha |\geq t\Rightarrow B\subseteq \alpha N}$.

5. B is absorbed by every balanced neighborhood of the origin,[1]
6. for every sequence of scalars ${\displaystyle \left(s_{i}\right)_{i=1}^{\infty }}$ that converges to 0 and every sequence ${\displaystyle \left(b_{i}\right)_{i=1}^{\infty }}$ in B, the sequence ${\displaystyle \left(s_{i}b_{i}\right)_{i=1}^{\infty }}$ converges to 0 in X.[1]
   • This was the definition of "bounded" that Kolmogorov used in 1934, which is the same as the definition introduced by Mazur and 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. every countable subset of B is bounded (according to any defining condition other than this one).[1]

and 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:

8. p(B) is bounded for all p ∈ 𝒫.[1]
9. there exists a sequence of non-0 scalars ${\displaystyle \left(s_{i}\right)_{i=1}^{\infty }}$ such that for every sequence ${\displaystyle \left(b_{i}\right)_{i=1}^{\infty }}$ in B, the sequence ${\displaystyle \left(s_{i}b_{i}\right)_{i=1}^{\infty }}$ is bounded in X (according to any defining condition other than this one).[1]
10. for all p ∈ 𝒫 , B is bounded (according to any defining condition other than this one) in the semi normed space (X, p).

and 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:

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

A vector subspace M of a TVS X is bounded if and only if M is contained in the closure of { 0 }.[1]

The collection of all bounded sets on a topological vector space X is called the bornology of X.

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

• The closure and interior of a bounded set is bouned.[1]
• Finite unions, finite sums, scalar multiples, and subsets of bounded sets are again 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.
• In any TVS, the balanced hull of a bounded set is again bounded.[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 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]

• In any topological vector space (TVS), finite sets are bounded.[1]
• Every totally bounded subset of a TVS is bounded.[1]
• The set of points of a Cauchy sequence is bounded, the set of points of a Cauchy net need not be bounded.

• The closure of a bounded set is bounded.
• In a locally convex space, the convex envelope of a bounded set is bounded. (Without local convexity this is false, as the ${\displaystyle L^{p}}$ spaces for ${\displaystyle 0<p<1}$ have no nontrivial open convex subsets.)
• The finite union or finite sum of bounded sets is bounded.
• Continuous linear mappings between topological vector spaces preserve boundedness.
• 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.
• A set A is bounded if and only if every countable subset of A is bounded

Theorem (Mackey's countability condition):[1] Suppose that X is a metrizable locally convex TVS and that ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$ is a countable sequence of bounded subsets of X. Then there exists a bounded subset B of X and a sequence ${\displaystyle \left(s_{i}\right)_{i=1}^{\infty }}$ of positive real numbers such that B_i ⊆ s_i B for all i.

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.

• Totally bounded space
• Local boundedness
• Bounded function
• Bounding point

• 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.