Bounded set (topological vector space)

For any set ${\displaystyle A}$ and scalar ${\displaystyle s,}$ let ${\displaystyle sA:=\{sa:a\in A\}.}$

Given a topological vector space (TVS) ${\displaystyle (X,\tau )}$ over a field ${\displaystyle \mathbb {K} ,}$ a subset ${\displaystyle B}$ of ${\displaystyle X}$ is called von Neumann bounded or just bounded in ${\displaystyle X}$ if any of the following equivalent conditions are satisfied:

1. Definition: For every neighborhood ${\displaystyle V}$ of the origin there exists a real ${\displaystyle r>0}$ such that ${\displaystyle B\subseteq sV}$ for all scalars ${\displaystyle s}$ satisfying ${\displaystyle |s|\geq r.}$[1]
   • This was the definition introduced by John von Neumann in 1935.[1]
2. ${\displaystyle B}$ is absorbed by every neighborhood of the origin.[2]
3. For every neighborhood ${\displaystyle V}$ of the origin there exists a scalar ${\displaystyle s}$ such that ${\displaystyle B\subseteq sV.}$
4. For every neighborhood ${\displaystyle V}$ of the origin there exists a real ${\displaystyle r>0}$ such that ${\displaystyle sB\subseteq V}$ for all scalars ${\displaystyle s}$ satisfying ${\displaystyle |s|\leq r.}$[1]
5. For every neighborhood ${\displaystyle V}$ of the origin there exists a real ${\displaystyle r>0}$ such that ${\displaystyle tB\subseteq V}$ for all real ${\displaystyle 0<t\leq r.}$[3]
6. 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: ${\displaystyle B}$ is bounded if and only if ${\displaystyle B}$ is absorbed by every balanced neighborhood of the origin.[1]
7. 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 ${\displaystyle B,}$ the sequence ${\displaystyle \left(s_{i}b_{i}\right)_{i=1}^{\infty }}$ converges to 0 in ${\displaystyle X.}$[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 the origin.[1]
8. For every sequence ${\displaystyle \left(b_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle B,}$ the sequence ${\displaystyle \left({\frac {1}{i}}b_{i}\right)_{i=1}^{\infty }\to 0}$ in ${\displaystyle X.}$[4]
9. Every countable subset of ${\displaystyle B}$ is bounded (according to any defining condition other than this one).[1]

while if ${\displaystyle X}$ is a locally convex space whose topology is defined by a family ${\displaystyle {\mathcal {P}}}$ of continuous seminorms, then this list may be extended to include:

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

while if ${\displaystyle X}$ is a seminormed space with seminorm ${\displaystyle p}$ (note that every normed space is a seminormed space and every norm is a seminorm), then this list may be extended to include:

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

while if ${\displaystyle B}$ is a vector subspace of the TVS ${\displaystyle X}$ then this list may be extended to include:

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

A subset that is not bounded is called unbounded.

Unless indicated otherwise, a topological vector space (TVS) need not be Hausdorff nor locally convex.

• In any TVS, finite unions, finite sums, scalar multiples, translations, 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 ${\displaystyle M}$ is a vector subspace of a TVS ${\displaystyle X}$ and if ${\displaystyle S\subseteq M,}$ then ${\displaystyle S}$ is bounded in ${\displaystyle M}$ if and only if it is bounded in ${\displaystyle X.}$[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 ${\displaystyle \{0\}}$ is bounded.

• Finite unions, finite Minkowski 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 space ${\displaystyle L^{p}}$ spaces for ${\displaystyle 0<p<1}$ have no nontrivial open convex subsets.
• A locally convex topological vector 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 ${\displaystyle A}$ of a topological module ${\displaystyle M}$ over a topological ring ${\displaystyle R}$ is bounded if for any neighborhood ${\displaystyle N}$ of ${\displaystyle 0_{M}}$ there exists a neighborhood ${\displaystyle w}$ of ${\displaystyle 0_{R}}$ such that ${\displaystyle wA\subseteq B.}$

• Bornological space – Space where bounded operators are continuous
• Bornivorous set – A set that can absorb any bounded subset
• Bounded function – A mathematical function the set of whose values is bounded
• Bounded operator – Linear transformation between TVSs
• Bounding point – Mathematical concept related to subsets of vector spaces
• Compact space – Type of mathematical space
• Kolmogorov's normability criterion – Characterization of normable spaces
• Local boundedness
• Totally bounded space

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