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 set 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

is a topological vector space (TVS) over a topological field

is called or just if any of the following equivalent conditions are satisfied:

• This was the definition introduced by John von Neumann in 1935.

is absorbed by every neighborhood of the origin. Any one of statements (1) through (5) above but with the word "neighborhood" replaced by any of the following: "balanced neighborhood," "open balanced neighborhood," "closed balanced neighborhood," "open neighborhood," "closed neighborhood".

• is absorbed by every balanced neighborhood of the origin.

• is locally convex then the adjective "convex" may be also be added to any of these 5 replacements.

• 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.

Every countable is bounded (according to any defining condition other than this one).

is a neighborhood basis at the origin then this list may be extended to include:

is a locally convex of continuous seminorms, then this list may be extended to include: (according to any defining condition other than this one). is bounded (according to any defining condition other than this one) in the semi normed space B is weakly bounded, i.e. every continuous linear functional is bounded on B

is a normed space with norm (or more generally, if it is a seminormed space is merely a seminorm), then this list may be extended to include: is a norm bounded

• is a linear map is a bounded linear operator (which recall means that its operator norm

is a subset of some (open or closed) ball.

• This ball need not be centered at the origin, but its radius must (as usual) be positive and finite.

then this list may be extended to include:

• is a Hausdorff space

A subset that is not bounded is called .

Bornology and fundamental systems of bounded sets

is called the or the ()

A or

Examples

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

Examples and sufficient conditions

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

Finite sets are bounded. Every totally bounded subset of a TVS is bounded. 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.

Unbounded sets

A set that is not bounded is said to be unbounded.

is unbounded

There exists a Fréchet space is

Stability properties

In any TVS, finite unions, finite Minkowski sums, scalar multiples, translations, subsets, closures, interiors, and balanced hullss of bounded sets are again bounded. In any locally convex TVS, the convex hull (also called the convex envelope) of a bounded set is again bounded. However, this may be false if the space is not locally convex, as the (non-locally convex) Lp space have no nontrivial open convex subsets. The image of a bounded set under a continuous linear map is a bounded subset of the codomain. A subset of an arbitrary (Cartesian) product of TVSs is bounded if and only if its image under every coordinate projections is bounded.

Properties

A locally convex topological vector space has a bounded neighborhood of zero if and only if its topology can be defined by a seminorm.

The polar of a bounded set is an absolutely convex and absorbing set.

Using the definition of uniformly bounded sets given below, Mackey's countability condition are bounded subsets of a metrizable locally convex space are uniformly bounded. In words, given any countable family of bounded sets in a metrizable locally convex space, it is possible to scale each set by its own positive real so that they become uniformly bounded.

Generalizations

Uniformly bounded sets

A family of sets of subsets of a topological vector space is said to be In the case of a normed (or seminormed is norm bounded

is said to be if and only if their operator norms

Q.E.D.

be a balanced is a homeomorphism

as desired.

is a nonmeager subset of itself (as it is a Baire space by the Baire category theorem

as desired. Q.E.D.

Since every singleton subset is an equicontinuous set of continuous linear operators between two topological vector spaces (not necessarily Hausdorff or locally convex), then the orbit

Bounded subsets of topological modules

The definition of bounded sets can be generalized to topological modules over a topological ring

See also

References

Notes

Bibliography