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 topological field
A subset of is called or just in if any of the following equivalent conditions are satisfied: <ol> <li>: For every neighborhood of the origin there exists a real such that for all scalars satisfying
<li> is absorbed by every neighborhood of the origin.</li> <li>For every neighborhood of the origin there exists a scalar such that </li> <li>For every neighborhood of the origin there exists a real such that for all scalars satisfying </li> <li>For every neighborhood of the origin there exists a real such that for all real </li> <li>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".
<li>For every sequence of scalars that converges to and every sequence in the sequence converges to in
<li>For every sequence in the sequence converges to in </li> <li>Every countable subset of is bounded (according to any defining condition other than this one).</li> </ol>
If is a neighborhood basis for at the origin then this list may be extended to include: <ol start=10> <li>Any one of statements (1) through (5) above but with the neighborhoods limited to those belonging to
</ol>
If is a locally convex space whose topology is defined by a family of continuous seminorms, then this list may be extended to include: <ol start=11> <li> is bounded for all </li> <li>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).</li> <li>For all is bounded (according to any defining condition other than this one) in the semi normed space </li> <li>B is weakly bounded, i.e. every continuous linear functional is bounded on B</li> </ol>
If is a normed space with norm (or more generally, if it is a seminormed space and is merely a seminorm), then this list may be extended to include: <ol start=14> <li> is a norm bounded subset of By definition, this means that there exists a real number such that for all </li> <li>
<li> is a subset of some (open or closed) ball.
</ol>
If is a vector subspace of the TVS then this list may be extended to include: <ol start=17> <li> is contained in the closure of
</ol>
A subset that is not bounded is called .
The collection of all bounded sets on a topological vector space is called the or the ()
A or of is a set of bounded subsets of such that every bounded subset of is a subset of some 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.
Unless indicated otherwise, a topological vector space (TVS) need not be Hausdorff nor locally convex.
<ul> <li>Finite sets are bounded.</li> <li>Every totally bounded subset of a TVS is bounded.</li> <li>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.</li> <li>The set of points of a Cauchy sequence is bounded, the set of points of a Cauchy net need not be bounded.</li> <li>The closure of the origin (referring to the closure of the set ) is always a bounded closed vector subspace. This set is the unique largest (with respect to set inclusion ) bounded vector subspace of In particular, if is a bounded subset of then so is </li> </ul>
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 contained in the closure (in ) of any bounded subset of
<ul> <li>In any TVS, finite unions, finite Minkowski sums, scalar multiples, translations, subsets, closures, interiors, and balanced hulls of bounded sets are again bounded.</li> <li>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 spaces for have no nontrivial open convex subsets.</li> <li>The image of a bounded set under a continuous linear map is a bounded subset of the codomain.</li> <li>A subset of an arbitrary (Cartesian) product of TVSs is bounded if and only if its image under every coordinate projections is bounded.</li> <li>If and is a topological vector subspace of then is bounded in if and only if is bounded in
</ul>
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 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 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 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
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.
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
Notes