# Bounded set (topological vector space)

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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.

## Contents

Bounded sets are a natural way to define 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

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]
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 [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:

1. ${\displaystyle p(B)}$ is bounded for all ${\displaystyle p\in {\mathcal {P}}.}$ [1]
2. 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]
3. 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:

1. 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:

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

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 ${\displaystyle X}$ is called the von Neumann bornology or the (canonical) bornology of ${\displaystyle X.}$

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

#### Examples

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

## Stability properties

Let ${\displaystyle X}$ 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 ${\displaystyle M}$ 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]

## Examples and sufficient conditions

### Non-examples

• In any TVS, any vector subspace that is not a contained in the closure of ${\displaystyle \{0\}}$ is unbounded (that is, not bounded).
• There exists a Fréchet space ${\displaystyle X}$ having a bounded subset ${\displaystyle B}$ and also a dense vector subspace ${\displaystyle M}$ such that ${\displaystyle B}$ is not contained in the closure (in ${\displaystyle X}$) of any bounded subset of ${\displaystyle M.}$ [5]

## Properties

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

## Generalization

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 ${\displaystyle wA\subseteq B.}$

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