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Vector bornology

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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.