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.
Suppose is 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:
If is a neighborhood basis for at the origin then this list may be extended to include:
If is a locally convex space whose topology is defined by a family of continuous seminorms, then this list may be extended to include:
If is a normed space with norm (or more generally, if it is a seminormed space and is merely a seminorm), [note 2] then this list may be extended to include:
If is a vector subspace of the TVS then this list may be extended to include:
A subset that is not bounded is called unbounded.
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
In any locally convex TVS, the set of closed and bounded disks are a base of bounded set. [1]
Unless indicated otherwise, a topological vector space (TVS) need not be Hausdorff nor locally convex.
Unbounded sets
A set that is not bounded is said to be unbounded.
Any vector subspace of a TVS that is not a contained in the closure of is unbounded
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]
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 [7] — If is a countable sequence of bounded subsets of a metrizable locally convex topological vector space then there exists a bounded subset of and a sequence of positive real numbers such that for all (or equivalently, such that ).
Using the definition of uniformly bounded sets given below, Mackey's countability condition can be restated as: If are bounded subsets of a metrizable locally convex space then there exists a sequence of positive real numbers such that 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.
A family of sets of subsets of a topological vector space is said to be uniformly bounded in if there exists some bounded subset of such that
which happens if and only if its union
is a bounded subset of In the case of a normed (or seminormed) space, a family is uniformly bounded if and only if its union is norm bounded, meaning that there exists some real such that for every or equivalently, if and only if
A set of maps from to is said to be uniformly bounded on a given set if the family is uniformly bounded in which by definition means that there exists some bounded subset of such that or equivalently, if and only if is a bounded subset of A set of linear maps between two normed (or seminormed) spaces and is uniformly bounded on some (or equivalently, every) open ball (and/or non-degenerate closed ball) in if and only if their operator norms are uniformly bounded; that is, if and only if
Proposition [8] — Let be a set of continuous linear operators between two topological vector spaces and and let be any bounded subset of Then is uniformly bounded on (that is, the family is uniformly bounded in ) if any of the following conditions are satisfied:
Proof of part (1) [8] |
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Assume is equicontinuous and let be a neighborhood of the origin in Since is equicontinuous, there exists a neighborhood of the origin in such that for every Because is bounded in there exists some real such that if then So for every and every which implies that Thus is bounded in Q.E.D. |
Proof of part (2) [9] |
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Let be a balanced neighborhood of the origin in and let be a closed balanced neighborhood of the origin in such that Define which is a closed subset of (since is closed while every is continuous) that satisfies for every Note that for every non-zero scalar the set is closed in (since scalar multiplication by is a homeomorphism) and so every is closed in It will now be shown that from which follows. If then being bounded guarantees the existence of some positive integer such that where the linearity of every now implies thus and hence as desired. Thus expresses as a countable union of closed (in ) sets. Since is a nonmeager subset of itself (as it is a Baire space by the Baire category theorem), this is only possible if there is some integer such that has non-empty interior in Let be any point belonging to this open subset of Let be any balanced open neighborhood of the origin in such that The sets form an increasing (meaning implies ) cover of the compact space so there exists some such that (and thus ). It will be shown that for every thus demonstrating that is uniformly bounded in and completing the proof. So fix and Let The convexity of guarantees and moreover, since Thus which is a subset of Since is balanced and we have which combined with gives Finally, and imply as desired. Q.E.D. |
Since every singleton subset of is also a bounded subset, it follows that if is an equicontinuous set of continuous linear operators between two topological vector spaces and (not necessarily Hausdorff or locally convex), then the orbit of every is a bounded subset of
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
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Notes