In mathematics, specifically in functional analysis, a family of subsets a topological vector space (TVS) is said to be saturated if contains a non-empty subset of and if for every the following conditions all hold:
contains every subset of ;
the union of any finite collection of elements of is an element of ;
If is any collection of subsets of then the smallest saturated family containing is called the saturated hull of [1]
The family is said to cover if the union is equal to ; it is total if the linear span of this set is a dense subset of [1]
Examplesedit
The intersection of an arbitrary family of saturated families is a saturated family.[1] Since the power set of is saturated, any given non-empty family of subsets of containing at least one non-empty set, the saturated hull of is well-defined.[2] Note that a saturated family of subsets of that covers is a bornology on
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN978-1584888666. 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. ISBN978-1-4612-7155-0. OCLC 840278135.
saturated, family, mathematics, specifically, functional, analysis, family, displaystyle, mathcal, subsets, topological, vector, space, displaystyle, said, saturated, displaystyle, mathcal, contains, empty, subset, displaystyle, every, displaystyle, mathcal, f. In mathematics specifically in functional analysis a family G displaystyle mathcal G of subsets a topological vector space TVS X displaystyle X is said to be saturated if G displaystyle mathcal G contains a non empty subset of X displaystyle X and if for every G G displaystyle G in mathcal G the following conditions all hold G displaystyle mathcal G contains every subset of G displaystyle G the union of any finite collection of elements of G displaystyle mathcal G is an element of G displaystyle mathcal G for every scalar a displaystyle a G displaystyle mathcal G contains aG displaystyle aG the closed convex balanced hull of G displaystyle G belongs to G displaystyle mathcal G 1 Contents 1 Definitions 2 Examples 3 See also 4 ReferencesDefinitions editIf S displaystyle mathcal S nbsp is any collection of subsets of X displaystyle X nbsp then the smallest saturated family containing S displaystyle mathcal S nbsp is called the saturated hull of S displaystyle mathcal S nbsp 1 The family G displaystyle mathcal G nbsp is said to cover X displaystyle X nbsp if the union G GG displaystyle bigcup G in mathcal G G nbsp is equal to X displaystyle X nbsp it is total if the linear span of this set is a dense subset of X displaystyle X nbsp 1 Examples editThe intersection of an arbitrary family of saturated families is a saturated family 1 Since the power set of X displaystyle X nbsp is saturated any given non empty family G displaystyle mathcal G nbsp of subsets of X displaystyle X nbsp containing at least one non empty set the saturated hull of G displaystyle mathcal G nbsp is well defined 2 Note that a saturated family of subsets of X displaystyle X nbsp that covers X displaystyle X nbsp is a bornology on X displaystyle X nbsp The set of all bounded subsets of a topological vector space is a saturated family See also editTopology of uniform convergence Topological vector lattice Vector lattice Partially ordered vector space ordered as a latticePages displaying short descriptions of redirect targetsReferences edit a b c d Schaefer amp Wolff 1999 pp 79 82 Schaefer amp Wolff 1999 pp 79 88 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 Treves Francois 2006 1967 Topological Vector Spaces Distributions and Kernels Mineola N Y Dover Publications ISBN 978 0 486 45352 1 OCLC 853623322 Retrieved from https en wikipedia org w index php title Saturated family amp oldid 1108338343, wikipedia, wiki, book, books, library,