Bounded set (topological vector space)

Summary

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 edit

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:

  1. Definition: For every neighborhood   of the origin there exists a real   such that  [note 1] 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 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".
    • e.g. Statement (2) may become:   is bounded if and only if   is absorbed by every balanced neighborhood of the origin.[1]
    • If   is locally convex then the adjective "convex" may be also be added to any of these 5 replacements.
  7. For every sequence of scalars   that converges to   and every sequence   in   the sequence   converges to   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   converges to   in  [4]
  9. Every countable subset of   is bounded (according to any defining condition other than this one).[1]

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

  1. Any one of statements (1) through (5) above but with the neighborhoods limited to those belonging to  
    • e.g. Statement (3) may become: For every   there exists a scalar   such that  

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  
  4. B is weakly bounded, i.e. every continuous linear functional is bounded on B[5]

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:

  1.   is a norm bounded subset of   By definition, this means that there exists a real number   such that   for all  [1]
  2.  
    • Thus, if   is a linear map between two normed (or seminormed) spaces and if   is the closed (alternatively, open) unit ball in   centered at the origin, then   is a bounded linear operator (which recall means that its operator norm   is finite) if and only if the image   of this ball under   is a norm bounded subset of  
  3.   is a subset of some (open or closed) ball.[note 3]
    • This ball need not be centered at the origin, but its radius must (as usual) be positive and finite.

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

  1.   is contained in the closure of  [1]
    • In other words, a vector subspace of   is bounded if and only if it is a subset of (the vector space)  
    • Recall that   is a Hausdorff space if and only if   is closed in   So the only bounded vector subspace of a Hausdorff TVS is  

A subset that is not bounded is called unbounded.

Bornology and fundamental systems of bounded sets edit

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 edit

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

Examples and sufficient conditions edit

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

  • 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.
  • 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  

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  [6]

Stability properties edit

  • In any TVS, finite unions, finite Minkowski sums, scalar multiples, translations, subsets, closures, interiors, and balanced hulls of bounded sets are again bounded.[1]
  • In any locally convex TVS, the convex hull (also called the convex envelope) of a bounded set is again bounded.[7] 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.[7]
  • The image of a bounded set under a continuous linear map is a bounded subset of the codomain.[1]
  • A subset of an arbitrary (Cartesian) product of TVSs is bounded if and only if its image under every coordinate projections is bounded.
  • If   and   is a topological vector subspace of   then   is bounded in   if and only if   is bounded in  [1]
    • In other words, a subset   is bounded in   if and only if it is bounded in every (or equivalently, in some) topological vector superspace of  

Properties edit

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[8] — 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.

Generalizations edit

Uniformly bounded sets edit

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[9] — 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:

  1.   is equicontinuous.
  2.   is a convex compact Hausdorff subspace of   and for every   the orbit   is a bounded subset of  
Proof of part (1)[9]

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)[10]

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  

Bounded subsets of topological modules edit

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 edit

References edit

  1. ^ a b c d e f g h i j k l m n o p q r Narici & Beckenstein 2011, pp. 156–175.
  2. ^ Schaefer 1970, p. 25.
  3. ^ Rudin 1991, p. 8.
  4. ^ Wilansky 2013, p. 47.
  5. ^ Narici Beckenstein (2011). Topological Vector Spaces (2nd ed.). pp. 253, Theorem 8.8.7. ISBN 978-1-58488-866-6.
  6. ^ Wilansky 2013, p. 57.
  7. ^ a b Narici & Beckenstein 2011, p. 162.
  8. ^ Narici & Beckenstein 2011, p. 174.
  9. ^ a b Rudin 1991, pp. 42−47.
  10. ^ Rudin 1991, pp. 46−47.

Notes

  1. ^ For any set   and scalar   the notation   is denotes the set  
  2. ^ This means that the topology on   is equal to the topology induced on it by   Note that every normed space is a seminormed space and every norm is a seminorm. The definition of the topology induced by a seminorm is identical to the definition of the topology induced by a norm.
  3. ^ If   is a normed space or a seminormed space, then the open and closed balls of radius   (where   is a real number) centered at a point   are, respectively, the sets   and   Any such set is called a (non-degenerate) ball.

Bibliography edit

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  • Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
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