# 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*:*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:

- 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]}

- This was the definition introduced by John von Neumann in 1935.
- B is absorbed by every neighborhood of the origin;
^{[2]} - for every neighborhood V of the origin there exists a scalar s such that
*B*⊆*sV*; - 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]} - 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]}

- e.g. Condition 2 may become: B is bounded if and only if B is absorbed by every balanced neighborhood of the origin.
- 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]}

- 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.
- for every sequence (
*b*_{n})^{∞}_{n=1}in B, the sequence (1/*n**b*_{n})^{∞}_{n=1}→ 0 in X;^{[3]} - 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:

*p*(*B*) is bounded for all*p*∈ 𝒫.^{[1]}- 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*_{i}*b*_{i})^{∞}_{i=1}is bounded in X (according to any defining condition other than this one).^{[1]} - 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:

- 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:

- 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 ℬ 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
*S*⊆*M*, then S is bounded in M if and only if it is bounded in X.^{[1]}

## Examples and sufficient conditions

- 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.

### 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 L
^{p}spaces for 0 <*p*< 1 have no nontrivial open convex subsets.

- Without local convexity this is false, as the L
- 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 (*B*_{i})^{∞}_{i=1} is a countable sequence of bounded subsets of X.
Then there exists a bounded subset B of X and a sequence (*r*_{i})^{∞}_{i=1} of positive real numbers such that *B*_{i} ⊆ *r*_{i} *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 *0 _{M}* there exists a neighborhood w of 0

_{R}such that

*w A ⊂ N*.

## See also

- 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

## References

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