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 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 and scalar let

Given a topological vector space (TVS) over a field a subset of is called ** von Neumann bounded** or just

**in if any of the following equivalent conditions are satisfied:**

*bounded**Definition*: For every neighborhood of the origin there exists a real such that for all scalars satisfying^{[1]}- This was the definition introduced by John von Neumann in 1935.
^{[1]}

- This was the definition introduced by John von Neumann in 1935.
- is absorbed by every neighborhood of the origin.
^{[2]} - For every neighborhood of the origin there exists a scalar such that
- For every neighborhood of the origin there exists a real such that for all scalars satisfying
^{[1]} - For every neighborhood of the origin there exists a real such that for all real
^{[3]} - 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: is bounded if and only if is absorbed by every balanced neighborhood of the origin.
^{[1]}

- e.g. Condition 2 may become: is bounded if and only if is absorbed by every balanced neighborhood of the origin.
- For every sequence of scalars that converges to 0 and every sequence in the sequence converges to 0 in
^{[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]}

- 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.
- For every sequence in the sequence in
^{[4]} - Every countable subset of is bounded (according to any defining condition other than this one).
^{[1]}

while if is a locally convex space whose topology is defined by a family of continuous seminorms, then this list may be extended to include:

- is bounded for all
^{[1]} - There exists a sequence of non-zero scalars such that for every sequence in the sequence is bounded in (according to any defining condition other than this one).
^{[1]} - For all is bounded (according to any defining condition other than this one) in the semi normed space

while if is a seminormed space with seminorm (note that every normed space is a seminormed space and every norm is a seminorm), then this list may be extended to include:

- There exists a real that for all
^{[1]}

while if is a vector subspace of the TVS then this list may be extended to include:

- is contained in the closure of
^{[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 is called the *von Neumann bornology* or the (*canonical*) *bornology of *

A *base* or *fundamental system of bounded sets* of is a set of bounded subsets of such that every bounded subset of is a subset of some ^{[1]}
The set of all bounded subsets of trivially forms a fundamental system of bounded sets of

#### Examples

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

## Stability properties

Let 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 is a vector subspace of a TVS and if then is bounded in if and only if it is bounded in
^{[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 is bounded.

### Non-examples

- In any TVS, any vector subspace that is not a contained in the closure of is unbounded (that is,
**not**bounded). - There exists a Fréchet space having a bounded subset and also a dense vector subspace such that is
*not*contained in the closure (in ) of any bounded subset of^{[5]}

## Properties

- 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 spaces for 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.

**Mackey's countability condition** (^{[1]}) — Suppose that is a metrizable locally convex TVS and that is a countable sequence of bounded subsets of
Then there exists a bounded subset of and a sequence of positive real numbers such that for all

## Generalization

The definition of bounded sets can be generalized to topological modules. A subset of a topological module over a topological ring is bounded if for any neighborhood of there exists a neighborhood of such that

## See also

- Bornivorous set – A set that can absorb any bounded subset
- Bounded function – Type of mathematical function whose values are bounded
- Bounded operator – A linear operator that sends bounded subsets to bounded subsets
- Bounding point – Mathematical concept related to subsets of vector spaces
- 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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