Vector bornology

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

Definitions

Prerequisits
Main article: Bornology

A bornology on a set X is a collection ℬ of subsets of X such that

  1. ℬ covers X (i.e. X = );
  2. ℬ is stable under inclusion (i.e. if B ∈ ℬ then every subset of B belongs to ℬ);
  3. ℬ is stable under finite unions (or equivalently, the union of any two sets in ℬ also belongs to ℬ).

in which case the pair (X, ℬ) is called a bounded structure.[1] Elements of ℬ are called ℬ-bounded sets or simply bounded. A subset 𝒜 of a bornology ℬ is called a base or fundamental system of ℬ if for every B ∈ ℬ, there exists an A ∈ 𝒜 such that B ⊆ A. Given a collection 𝒮 of subsets of X, the smallest bornology containing 𝒮 is called the bornology generated by 𝒮.[1]

If (X, 𝒜) and (Y, ℬ) are bounded structures and f: X → Y is a map then f is called locally bounded or just bounded if the image under f of every 𝒜-bounded set is a ℬ-bounded set; that is, if for every A ∈ 𝒜, f(A) ∈ ℬ.[1]

If (X, 𝒜) and (Y, ℬ) are bounded structures then the product bornology on X × Y is the bornology having as a base the collection of all sets of the form A × B, where A ∈ 𝒜 and B ∈ ℬ.[1] One may show that a subset of X × Y is bounded in the product bornology if and only if its image under the canonical projections onto X and Y are both bounded.

Vector bornology

Let X be a vector space over a field 𝔽, where 𝔽 has a bornology ℬ𝔽. A bornology ℬ on X is called a vector bornology if its vector spaces operations of addition and scalar multiplication are bounded maps.[1] Explicitly, this means that when X is endowed with the bornology ℬ then:

  1. the addition map X × X → X defined by (x, y) ↦ x + y is a bounded map, where X × X has the product bornology, and
  2. the scalar multiplication map 𝔽 × X → X defined by (s, x) ↦ sx is a bounded map, where 𝔽 × X has the product bornology induced by (𝔽, ℬ𝔽) and (X, ℬ).

Usually, 𝔽 is either the real or complex numbers, in which case we call a vector bornology ℬ on X a convex vector bornology if ℬ has a base consisting of convex sets.

Characterizations

Suppose that X is a topological vector space (TVS) over the field 𝔽 of real or complex numbers and ℬ is a bornology on X. Then the following are equivalent:

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

Bornology on a topological vector space

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

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

Examples

Locally convex space of bounded functions

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

Bornology of equicontinuity

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

See also

Citations
  1. Narici & Beckenstein 2011, pp. 156–175.

Sources
  • Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co. pp. xii+144. ISBN 0-7204-0712-5. MR 0500064.
  • 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 978-158488866-6. OCLC 144216834.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Volume 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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