In mathematics, especially functional analysis, a bornology on a vector space over a field where has a bornology ℬ, is called a vector bornology if makes the vector space operations into bounded maps.

Definitions

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Prerequisits

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A bornology on a set   is a collection   of subsets of   that satisfy all the following conditions:

  1.   covers   that is,  
  2.   is stable under inclusions; that is, if   and   then  
  3.   is stable under finite unions; that is, if   then  

Elements of the collection   are called  -bounded or simply bounded sets if   is understood. The pair   is called a bounded structure or a bornological set.

A base or fundamental system of a bornology   is a subset   of   such that each element of   is a subset of some element of   Given a collection   of subsets of   the smallest bornology containing   is called the bornology generated by  [1]

If   and   are bornological sets then their product bornology on   is the bornology having as a base the collection of all sets of the form   where   and  [1] A subset of   is bounded in the product bornology if and only if its image under the canonical projections onto   and   are both bounded.

If   and   are bornological sets then a function   is said to be a locally bounded map or a bounded map (with respect to these bornologies) if it maps  -bounded subsets of   to  -bounded subsets of   that is, if  [1] If in addition   is a bijection and   is also bounded then   is called a bornological isomorphism.

Vector bornology

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Let   be a vector space over a field   where   has a bornology   A bornology   on   is called a vector bornology on   if it is stable under vector addition, scalar multiplication, and the formation of balanced hulls (i.e. if the sum of two bounded sets is bounded, etc.).

If   is a vector space and   is a bornology on   then the following are equivalent:

  1.   is a vector bornology
  2. Finite sums and balanced hulls of  -bounded sets are  -bounded[1]
  3. The scalar multiplication map   defined by   and the addition map   defined by   are both bounded when their domains carry their product bornologies (i.e. they map bounded subsets to bounded subsets)[1]

A vector bornology   is called a convex vector bornology if it is stable under the formation of convex hulls (i.e. the convex hull of a bounded set is bounded) then   And a vector bornology   is called separated if the only bounded vector subspace of   is the 0-dimensional trivial space  

Usually,   is either the real or complex numbers, in which case a vector bornology   on   will be called a convex vector bornology if   has a base consisting of convex sets.

Characterizations

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Suppose that   is a vector space over the field   of real or complex numbers and   is a bornology on   Then the following are equivalent:

  1.   is a vector bornology
  2. addition and scalar multiplication are bounded maps[1]
  3. the balanced hull of every element of   is an element of   and the sum of any two elements of   is again an element of  [1]

Bornology on a topological vector space

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If   is a topological vector space then the set of all bounded subsets of   from a vector bornology on   called the von Neumann bornology of  , the usual bornology, or simply the bornology of   and is referred to as natural boundedness.[1] In any locally convex topological vector space   the set of all closed bounded disks form a base for the usual bornology of  [1]

Unless indicated otherwise, it is always assumed that the real or complex numbers are endowed with the usual bornology.

Topology induced by a vector bornology

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Suppose that   is a vector space over the field   of real or complex numbers and   is a vector bornology on   Let   denote all those subsets   of   that are convex, balanced, and bornivorous. Then   forms a neighborhood basis at the origin for a locally convex topological vector space topology.

Examples

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Locally convex space of bounded functions

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Let   be the real or complex numbers (endowed with their usual bornologies), let   be a bounded structure, and let   denote the vector space of all locally bounded  -valued maps on   For every   let   for all   where this defines a seminorm on   The locally convex topological vector space topology on   defined by the family of seminorms   is called the topology of uniform convergence on bounded set.[1] This topology makes   into a complete space.[1]

Bornology of equicontinuity

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Let   be a topological space,   be the real or complex numbers, and let   denote the vector space of all continuous  -valued maps on   The set of all equicontinuous subsets of   forms a vector bornology on  [1]

See also

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Citations

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  1. ^ a b c d e f g h i j k l Narici & Beckenstein 2011, pp. 156–175.

Bibliography

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  • Hogbe-Nlend, Henri (1977). Bornologies and Functional Analysis: Introductory Course on the Theory of Duality Topology-Bornology and its use in Functional Analysis. North-Holland Mathematics Studies. Vol. 26. Amsterdam New York New York: North Holland. ISBN 978-0-08-087137-0. MR 0500064. OCLC 316549583.
  • Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. ISBN 978-082180780-4.
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.