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.

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

  1. Definition: For every neighborhood of the origin there exists a real such that for all scalars satisfying [1]
  2. is absorbed by every neighborhood of the origin. [2]
  3. For every neighborhood of the origin there exists a scalar such that
  4. For every neighborhood of the origin there exists a real such that for all scalars satisfying [1]
  5. For every neighborhood of the origin there exists a real such that for all real [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: is bounded if and only if is absorbed by every balanced neighborhood of the origin. [1]
  7. 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]
  8. For every sequence in the sequence in [4]
  9. 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:

  1. is bounded for all [1]
  2. 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]
  3. 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:

  1. 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:

  1. 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).

Examples and sufficient conditions

Non-examples

Properties

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

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