Book Section records
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 08 Dec 2023 12:24:26 +0000Some Examples of Normed Köthe Spaces
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Authors: Luxemburg, W. A. J.; Zaanen, A. C.
Year: 1966
DOI: 10.1007/978-3-642-85997-7_22
Let X be a non-empty point set, and μ a countably additive and non-negative measure in X. We assume that the Carathéodory extension procedure has already been applied to μ, so that the σ-field Λ on which μ is defined cannot be enlarged by another application of the Carathéodory procedure. Furthermore, it will be assumed that μ is (totally) (σ-finite, i.e., X is the union of a finite or countable number of sets of finite measure. Hence, the triple (X, Λ, μ) is a (totally) σ-finite measure space in the usual terminology. The notation ∫ d μ will denote integration (with respect to μ) over the whole set X, and χ E = χ E (x) will stand for the characteristic function of the set E ⊂ X.https://authors.library.caltech.edu/records/cvxy3-d6y83Closure properties of sequences of exponentals { exp (i λ_n t) }
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Authors: Luxemburg, W. A. J.
Year: 1974
DOI: 10.1007/bfb0064735
[no abstract]https://authors.library.caltech.edu/records/p4w8d-p7k62Non-Standard Analysis
https://resolver.caltech.edu/CaltechAUTHORS:20200519-134358780
Authors: Luxemburg, W. A. J.
Year: 1977
DOI: 10.1007/978-94-010-1138-9_6
1. As early as 1934 it was pointed out by Thoralf Skolem (see [17]) that there exist proper extensions of the natural number system which have, in some sense, 'the same properties' as the natural numbers. The title of Skolem's paper indicates that the purpose of it was to show that no axiomatic system specified in a formal language, in Skolem's case the lower predicate calculus, can characterize the natural numbers categorically. At that time, however, Skolem did not concern himself with the properties of the structures whose existence he had established. In due course these structures became known as non-standard models of arithmetic. For nearly thirty years since the appearance of Skolem's paper non-standard models were not used or considered in any sense by the working mathematician. Robinson's fundamental paper, which appeared in 1961 under the title 'Non-standard Analysis', (see [11]) changed this situation dramatically. In this paper Abraham Robinson was the first to point out that this highly abstract part of model theory could be applied fruitfully to a theory so far removed from it as the infinitesimal calculus. As a result Robinson obtained a firm foundation for the non-archimedian approach to the calculus based on a number system containing infinitely small and infinitely large numbers, in a manner almost identical to that suggested by Leibniz some three centuries ago, and which predominated the calculus until the middle of the nineteenth century when it was rejected as unsound and replaced by the ϵ, δ-method of Weierstrass.https://authors.library.caltech.edu/records/rqbsj-s9010Robinson, Abraham (1918–1974)
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Authors: Luxemburg, W. A. J.
Year: 1987
DOI: 10.1057/978-1-349-95121-5_1579-1
A logician, mathematician and applied mathematician, Abraham Robinson was one of the foremost proponents of applying the methods and results of mathematical logic, in particular model theory to mathematics. This point of view led Abraham Robinson around 1960 to the creation of Non-standard Analysis.https://authors.library.caltech.edu/records/65e8h-mfr78Robinson's Nonstandard Analysis and its Influence on Mathematics
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Authors: Luxemburg, Wilhelmus A. J.
Year: 1990
DOI: 10.1007/978-3-0348-5242-5_2
The theme of the Tagung concerns the question: Resultatismus oder Geneseologie? Impulse der Nichtstandard Analysis für die Geschichtsforschung.https://authors.library.caltech.edu/records/349kw-39839Integration with Respect to Finitely Additive Measures
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155855248
Authors: Luxemburg, Wilhelmus A. J.
Year: 1991
DOI: 10.1007/978-3-642-58199-1_6
This essay interprets the theory of finitely additive measures within the framework of the theory of Riesz spaces. The following topics are discussed: the extension procedures of measures, the Riemann and the Dunford integration procedures, the Radon-Nikodym Theorem and the Hahn Decomposition Theorem, the representation theory of the Radon- Nikodym derivatives as generalized functions, conditional expectation operators, the theory of L^p -spaces, and the norm completeness problem.
The nature of the classical axiom of countable additivity is examined from Carathéodory's algebraic measure-theoretic point of view.https://authors.library.caltech.edu/records/bmdz0-mrg72An Alternative Proof of a Radon-Nikodym Theorem for Lattice Homomorphisms
https://resolver.caltech.edu/CaltechAUTHORS:20200515-115519213
Authors: Huijsmans, C. B.; Luxemburg, W. A. J.
Year: 1992
DOI: 10.1007/978-94-017-2721-1_7
We give a new proof of the Luxemburg-Schep theorem for lattice homomorphisms.https://authors.library.caltech.edu/records/tpkdd-f1132Diagonals of the Powers of an Operator on a Banach Lattice
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Authors: Luxemburg, W. A. J.; de Pagter, B.; Schep, A. R.
Year: 1995
DOI: 10.1007/978-3-0348-9076-2_13
This paper is devoted to a detailed study of the properties of the band projection D of the complete lattice ordered algebra L_r(E) of the regular (or order bounded)
operators of a Dedekind complete Banach lattice E onto the center Z(E) of E. We recall that the center Z(E) is the commutative subalgebra of L_r(E) of all T satisfying |T| ≤ λI where I is the identity operator. In the finite dimensional case, with respect to the standard numerical basis, Z(E) is the algebra of all diagonal matrices. For this reason the band projection D is called the diagonal map of E.https://authors.library.caltech.edu/records/fqayp-zj228Biographical Notes
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Authors: Huijsmans, C. B.; Kaashoek, M. A.; Luxemburg, W. A. J.; de Pagter, B.
Year: 1995
DOI: 10.1007/978-3-0348-9076-2_1
During the first week of September 1993, a Symposium was held at the University of Leiden honoring Professor A.C. Zaanen on the occasion of his 80th birthday in June of the same year. In March 1993, Professor Zaanen also celebrated the 55th anniversary of receiving his Doctor's Degree in Philosophy at the University of Leiden, marking the beginning of his remarkable mathematical career.https://authors.library.caltech.edu/records/7g4qz-srh26Robinson, Abraham (1918–1974)
https://resolver.caltech.edu/CaltechAUTHORS:20210203-081026434
Authors: Luxemburg, W. A. J.
Year: 2018
DOI: 10.1057/978-1-349-95189-5_1579
A logician, mathematician and applied mathematician, Abraham Robinson was one of the foremost proponents of applying the methods and results of mathematical logic, in particular model theory to mathematics. This point of view led Abraham Robinson around 1960 to the creation of Non-standard Analysis.https://authors.library.caltech.edu/records/n9wt3-s4178