Combined Feed
https://feeds.library.caltech.edu/people/Luxemburg-W-A-J/combined.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 08 Dec 2023 21:55:14 +0000Reflexivity of the length function
https://resolver.caltech.edu/CaltechAUTHORS:20181009-151135685
Authors: Halperin, Israel; Luxemburg, W. A. J.
Year: 1957
DOI: 10.1090/S0002-9939-1957-0087903-X
[no abstract]https://authors.library.caltech.edu/records/ebern-y4c21On the convergence of successive approximations in the theory of ordinary differential equations
https://resolver.caltech.edu/CaltechAUTHORS:20181009-150532893
Authors: Luxemburg, W. A. J.
Year: 1958
DOI: 10.4153/CMB-1958-003-5
[no abstract]https://authors.library.caltech.edu/records/w856h-y4v13Numerical methods and existence theorems for ordinary differential equations
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854757
Authors: Hull, T. E.; Luxemburg, W. A. J.
Year: 1960
DOI: 10.1007/bf01386206
[no abstract]https://authors.library.caltech.edu/records/pv2ek-9jy26A remark on R. R. PHELPS' paper "subreflexive normed linear spaces"
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854862
Authors: Luxemburg, W. A. J.
Year: 1960
DOI: 10.1007/bf01236931
In a recent paper on subreflexive normed linear spaces (see Arch. Math. 8, 444--450 (1957)) and its correction (see Arch. Math. 9,439--440, (1958)) R. R. PHELPS proved the following theorem.https://authors.library.caltech.edu/records/mnkcz-9bc29Two applications of the method of construction by ultrapowers to anaylsis
https://resolver.caltech.edu/CaltechAUTHORS:20181009-145540187
Authors: Luxemburg, W. A. J.
Year: 1962
DOI: 10.1090/S0002-9904-1962-10824-6
[no abstract]https://authors.library.caltech.edu/records/24mm1-zbt22A Property of the Fourier Coefficients of an Integrable Function
https://resolver.caltech.edu/CaltechAUTHORS:20181009-145207989
Authors: Luxemburg, W. A. J.
Year: 1962
DOI: 10.2307/2312535
[no abstract]https://authors.library.caltech.edu/records/5cgv5-17k04Compactness of integral operators in Banach function spaces
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854668
Authors: Luxemburg, W. A. J.; Zaanen, A. C.
Year: 1963
DOI: 10.1007/bf01349240
[no abstract]https://authors.library.caltech.edu/records/m9pqb-ztc94On Finitely Additive Measures in Boolean Algebras
https://resolver.caltech.edu/CaltechAUTHORS:20181009-144613726
Authors: Luxemburg, W. A. J.
Year: 1964
DOI: 10.1515/crll.1964.213.165
[no abstract]https://authors.library.caltech.edu/records/5k5g6-hdy79A remark on Sikorski's extension theorem for homomorphisms in the theory of Boolean algebras
https://resolver.caltech.edu/CaltechAUTHORS:20181009-143713456
Authors: Luxemburg, W. A. J.
Year: 1964
DOI: 10.4064/fm-55-3-239-247
[no abstract]https://authors.library.caltech.edu/records/a37xm-vdm93Some Examples of Normed Köthe Spaces
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854393
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-d6y83Some examples of normed Köthe spaces
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854579
Authors: Luxemburg, W. A. J.; Zaanen, A. C.
Year: 1966
DOI: 10.1007/bf01369107
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/p7b7q-dwk51Archimedean quotient Riesz spaces
https://resolver.caltech.edu/CaltechAUTHORS:20181009-141551161
Authors: Luxemburg, W. A. J.; Moore Jr., L. C., Jr.
Year: 1967
DOI: 10.1215/S0012-7094-67-03475-8
[no abstract]https://authors.library.caltech.edu/records/nj55q-mkn11An extension of the concept of the order dual of a Riesz space
https://resolver.caltech.edu/CaltechAUTHORS:20181009-142824090
Authors: Luxemburg, W. A. J.; Masterson, J. J.
Year: 1967
DOI: 10.4153/CJM-1967-041-6
[no abstract]https://authors.library.caltech.edu/records/e8svt-2q535Is every integral normal?
https://resolver.caltech.edu/CaltechAUTHORS:20181009-143334534
Authors: Luxemburg, W. A. J.
Year: 1967
DOI: 10.1090/S0002-9904-1967-11825-1
[no abstract]https://authors.library.caltech.edu/records/133nf-08b65On some order properties of Riesz spaces and their relations
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854959
Authors: Luxemburg, W. A. J.
Year: 1968
DOI: 10.1007/bf01898770
The purpose of this note is to show that a number of order properties which occur in the theory of Riesz spaces are in fact equivalent. For a proper understanding of the kind of results we are interested in we shall first begin by recalling the various concepts which are involved.https://authors.library.caltech.edu/records/jyrkc-9sf89Entire Functions and Muntz-Szasz Type Approximation
https://resolver.caltech.edu/CaltechAUTHORS:20181009-141126373
Authors: Luxemburg, W. A. J.; Korevaar, J.
Year: 1971
DOI: 10.2307/1995828
Let [a, b] be a bounded interval with a>O. Under what conditions on
the sequence of exponents {A,,} can every function in LP[a, b] or C[a, b] be approxi mated arbitrarily closely by linear combinations of powers xAn? What is the distance between xA and the closed span Sc(xAn)? What is this closed span if not the whole space? Starting with the case of L2, C. H. Muntz and 0. Szasz considered the first two questions for the interval [0, 1]. L. Schwartz, J. A. Clarkson and P. Erdos, and the second author answered the third question for [0, 1] and also considered the interval [a, b]. For the case of [0, 1], L. Schwartz (and, earlier, in a limited way, T. Carleman) successfully used methods of complex and functional analysis, but until now the case of [a, b] had proved resistant to a direct approach of that kind. In the present paper complex analysis is used to obtain a simple direct treatment for the case of [a, b]. The crucial step is the construction of entire functions of exponential type which vanish at prescribed points not too close to the real axis and which, in a sense, are as small on both halves of the real axis as such functions can be. Under suitable conditions on the sequence of complex numbers {An} the construction leads readily to asymptotic lower bounds for the distances dk=d{xAk, Sc(xAn, nAk)}. These bounds are used to determine Sc(xAn) and to generalize a result for a boundary value problem for the heat equation obtained recently by V. J. Mizel and T. I. Seidman.https://authors.library.caltech.edu/records/k8dpn-thx56Arzela's Dominated Convergence Theorem for the Riemann Integral
https://resolver.caltech.edu/CaltechAUTHORS:20181009-140756074
Authors: Luxemburg, W. A. J.
Year: 1971
DOI: 10.2307/2317801
[no abstract]https://authors.library.caltech.edu/records/kvwyq-waq50On an inequality of A. Khintchine for zero-one matrices
https://resolver.caltech.edu/CaltechAUTHORS:20181009-140410509
Authors: Luxemburg, W. A. J.
Year: 1972
DOI: 10.1016/0097-3165(72)90043-X
Let A be a matrix of m rows and n columns whose entries are either zero or one with row i of sum ri (i = 1, 2,…, m) and column j of sum sj (j = 1, 2,…, n). Then a result of Khintchine states that , where l = max(m, n) and σ is the total number of ones in A. In the present paper a new proof of Khintchine's inequality is presented and a number of extensions to bounded plane measurable sets are discussed.https://authors.library.caltech.edu/records/02c8y-pek72What is nonstandard analysis?
https://resolver.caltech.edu/CaltechAUTHORS:20181009-135810847
Authors: Luxemburg, W. A. J.
Year: 1973
DOI: 10.2307/3038221
[no abstract]https://authors.library.caltech.edu/records/gxhx2-70092Closure properties of sequences of exponentals { exp (i λ_n t) }
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155855059
Authors: Luxemburg, W. A. J.
Year: 1974
DOI: 10.1007/bfb0064735
[no abstract]https://authors.library.caltech.edu/records/p4w8d-p7k62On an infinite series of Abel occurring in the theory of interpolation
https://resolver.caltech.edu/CaltechAUTHORS:20181009-135412183
Authors: Luxemburg, W. A. J.
Year: 1975
DOI: 10.1016/0021-9045(75)90020-9
The purpose of this paper is to show that for a certain class of functions f which are analytic in the complex plane possibly minus (−∞, −1], the Abel series ! is convergent for all β>0. Its sum is an entire function of exponential type and can be evaluated in terms of f. Furthermore, it is shown that the Abel series of f for small β>0 approximates f uniformly in half-planes of the form Re(z) ⩾ − 1 + δ, δ>0. At the end of the paper some special cases are discussed.https://authors.library.caltech.edu/records/4vzvm-e4c14On a class of valuation fields introduced by A. Robinson
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854296
Authors: Luxemburg, W. A. J.
Year: 1976
DOI: 10.1007/bf02756999
It is shown that the nonarchimedean valuation fields ^ρR introduced by A. Robinson are not only complete but are also spherically complete. Further-more, it is shown that to every normed linear space over the reals there exists a nonarchimedean normed linear space ^ρE over ^ρR in the sense of Monna which is spherically complete and extends E.https://authors.library.caltech.edu/records/e66gg-9q404Non-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)
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854487
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-mfr78The Work of Dorothy Maharam on Kernel Representations of Linear Operators
https://resolver.caltech.edu/CaltechAUTHORS:20181009-132515650
Authors: Luxemburg, W. A. J.
Year: 1989
DOI: 10.1090/conm/094/1012988
[no abstract]https://authors.library.caltech.edu/records/g19p0-n7c52Robinson's Nonstandard Analysis and its Influence on Mathematics
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155855154
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-39839Positive Operators, Riesz Spaces, and Economics
https://resolver.caltech.edu/CaltechAUTHORS:20181011-142851140
Authors: Aliprantis, Charalambos D.; Border, Kim C.; Luxemburg, Wilhelmus A. J.
Year: 1991
DOI: 10.1007/978-3-642-58199-1
Over the last fifty years advanced mathematical tools have become an integral part in the development of modern economic theory. Economists continue to invoke sophisticated mathematical techniques and ideas in order to understand complex economic and social problems. In the last ten years the theory of Riesz spaces (vector lattices) has been successfully applied to economic theory. By now it is understood relatively well that the lattice structure of Riesz spaces can be employed to capture and interpret several economic notions. On April 16-20, 1990, a small conference on Riesz Spaces, Positive Opera tors, and their Applications to Economics took place at the California Institute of Technology. The purpose of the conference was to bring mathematicians special ized in Riesz Spaces and economists specialized in General Equilibrium together to exchange ideas and advance the interdisciplinary cooperation between math ematicians and economists. This volume is a collection of papers that represent the talks and discussions of the participants at the week-long conference. We take this opportunity to thank all the participants of the conference, especially those whose articles are contained in this volume. We also greatly ap preciate the financial support provided by the California Institute of Technology. In particular, we express our sincerest thanks to David Grether, John Ledyard, and David Wales for their support. Finally, we would like to thank Susan Davis, Victoria Mason, and Marge D'Elia who handled the delicate logistics for the smooth running of the conference.https://authors.library.caltech.edu/records/p6hyp-bgv06Positive Operators, Riesz Spaces, and Economics
https://resolver.caltech.edu/CaltechAUTHORS:20200515-130605911
Authors: Aliprantis, Charalambos D.; Border, Kim C.; Luxemburg, Wilhelmus A. J.
Year: 1991
DOI: 10.1007/978-3-642-58199-1
Over the last fifty years advanced mathematical tools have become an integral part in the development of modern economic theory. Economists continue to invoke sophisticated mathematical techniques and ideas in order to understand complex economic and social problems. In the last ten years the theory of Riesz spaces (vector lattices) has been successfully applied to economic theory. By now it is understood relatively well that the lattice structure of Riesz spaces can be employed to capture and interpret several economic notions. On April 16-20, 1990, a small conference on Riesz Spaces, Positive Opera tors, and their Applications to Economics took place at the California Institute of Technology. The purpose of the conference was to bring mathematicians special ized in Riesz Spaces and economists specialized in General Equilibrium together to exchange ideas and advance the interdisciplinary cooperation between math ematicians and economists. This volume is a collection of papers that represent the talks and discussions of the participants at the week-long conference. We take this opportunity to thank all the participants of the conference, especially those whose articles are contained in this volume. We also greatly ap preciate the financial support provided by the California Institute of Technology. In particular, we express our sincerest thanks to David Grether, John Ledyard, and David Wales for their support. Finally, we would like to thank Susan Davis, Victoria Mason, and Marge D'Elia who handled the delicate logistics for the smooth running of the confer ence.https://authors.library.caltech.edu/records/wcnfe-2bg85Integration 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-mrg72Positive Operators and Semigroups on Banach Lattices: Proceedings of a Caribbean Mathematics Foundation Conference 1990
https://resolver.caltech.edu/CaltechAUTHORS:20181011-162006203
Authors: Huijsmans, C. B.; Luxemburg, W. A. J.
Year: 1992
DOI: 10.1007/978-94-017-2721-1
During the last twenty-five years, the development of the theory of Banach lattices has stimulated new directions of research in the theory of positive operators and the theory of semigroups of positive operators. In particular, the recent investigations in the structure of the lattice ordered (Banach) algebra of the order bounded operators of a Banach lattice have led to many important results in the spectral theory of positive operators. The contributions contained in this volume were presented as lectures at a conference organized by the Caribbean Mathematics Foundation, and provide an overview of the present state of development of various areas of the theory of positive operators and their spectral properties.
This book will be of interest to analysts whose work involves positive matrices and positive operators.https://authors.library.caltech.edu/records/gnkjs-phb84An 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-f1132An Alternative Proof of a Radon-Nikodym Theorem for Lattice Homomorphisms
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155855350
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/82syf-ajk89Multiplicativity factors for seminorms. II
https://resolver.caltech.edu/CaltechAUTHORS:20181009-132007385
Authors: Arens, Richard; Goldberg, Moshe; Luxemburg, W. A. J.
Year: 1992
DOI: 10.1016/0022-247X(92)90026-A
Let S be a seminorm on an algebra . In this paper we study multiplicativity and quadrativity factors for S, i.e., constants μ > 0 and λ > 0 for which S(xy) ⩽ μS(x)S(y) and S(x2) ⩽ λS(x)2 for all x, y ∈ A. We begin by investigating quadrativity factors in terms of the kernel of S. We then turn to the question, under what conditions does S have multiplicativity factors if it has quadrativity factors? We show that if is commutative then quadrativity factors imply multiplicativity factors. We further show that in the noncommutative case there exist both proper seminorms and norms that have quadrativity factors but no multiplicativity factors.https://authors.library.caltech.edu/records/69hyp-jeq94Multiplicativity Factors for Orlicz Space Function Norms
https://resolver.caltech.edu/CaltechAUTHORS:20181009-131120934
Authors: Arens, Richard; Goldberg, Moshe; Luxemburg, W. A. J.
Year: 1993
DOI: 10.1006/jmaa.1993.1264
Let ρφ be a function norm defined by a Young function φ with respect to a measure space (T, Ω, m), and let Lφ be the Orlicz space determined by ρφ. If Lφ is an algebra, then a constant μ > 0 is called a multiplicativity factor for ρφ, if ρφ,(fg) ≤ μρφ(f) ρφ(g) for all f, g ∈ Lφ. The main objective of this paper is to give conditions under which Lφ is indeed an algebra, and to obtain in this case the best (least) multiplicativity factor for ρφ. The first of our principal results is that Lφ is an algebra if and only if or Our second main result states that if Lφ is an algebra and (T, Ω, m) is free of infinite atoms, then the best multiplicativity factor for ρφ is φ−1(1/minf if minf > 0, and x∞(φ) if minf = 0.https://authors.library.caltech.edu/records/dcnbx-9tc44Multiplicativity Factors for Function Norms
https://resolver.caltech.edu/CaltechAUTHORS:20181009-131548532
Authors: Arens, R.; Goldberg, M.; Luxemburg, W. A. J.
Year: 1993
DOI: 10.1006/jmaa.1993.1263
Let (T, Ω, m) be a measure space; let ρ be a function norm on = (T, Ω, m), the algebra of measurable functions on T; and let Lρ be the space {f ∈ : ρ(f) < ∞} modulo the null functions. If Lρ, is an algebra, then we call a constant μ > 0 a multiplicativity factor for ρ if ρ(fg) ≤ μρ(f) ρ(g) for all f, g ∈ Lρ. Similarly, λ > 0 is a quadrativity factor if ρ(f2) ≤ λρ(f)2 for all f. The main purpose of this paper is to give conditions under which Lρ, is indeed an algebra, and to obtain in this case the best (least) multiplicativity and quadrativity factors for ρ. The first of our two principal results is that if ρ is σ-subadditive, then Lρ is an algebra if and only if Lρ is contained in L∞. Our second main result is that if (T, Ω, m) is free of infinite atoms, ρ is σ-subadditive and saturated, and Lρ, is an algebra, then the multiplicativity and quadrativity factors for ρ coincide, and the best such factor is determined by sup{||f||∞: f ∈ Lρ, ρ(f) ≤ 1}.https://authors.library.caltech.edu/records/60ptd-2r168Operator Theory in Function Spaces and Banach Lattices: Essays dedicated to A.C. Zaanen on the occasion of his 80th birthday
https://resolver.caltech.edu/CaltechAUTHORS:20181011-162816470
Authors: Huijsmans, C. B.; Kaashoek, M. A.; Luxemburg, W. A. J.; de Pagter, B.
Year: 1995
DOI: 10.1007/978-3-0348-9076-2
This volume is dedicated to A.C. Zaanen, one of the pioneers of functional analysis, and eminent expert in modern integration theory and the theory of vector lattices, on the occasion of his 80th birthday. The book opens with biographical notes, including Zaanen's curriculum vitae and list of publications. It contains a selection of original research papers which cover a broad spectrum of topics about operators and semigroups of operators on Banach lattices, analysis in function spaces and integration theory. Special attention is paid to the spectral theory of operators on Banach lattices; in particular, to the one of positive operators. Classes of integral operators arising in systems theory, optimization and best approximation problems, and evolution equations are also discussed. The book will appeal to a wide range of readers engaged in pure and applied mathematics.https://authors.library.caltech.edu/records/yn8wc-nq866Diagonals of the Powers of an Operator on a Banach Lattice
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155855535
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
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854110
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-srh26Advances in Analysis, Probability and Mathematical Physics: Contributions of Nonstandard Analysis
https://resolver.caltech.edu/CaltechAUTHORS:20181012-074421607
Authors: Albeverio, Sergio A.; Luxemburg, Wilhelm A. J.; Wolff, Manfred P. H.
Year: 1995
DOI: 10.1007/978-94-015-8451-7
In 1961 Robinson introduced an entirely new version of the theory of infinitesimals, which he called 'Nonstandard analysis'. 'Nonstandard' here refers to the nature of new fields of numbers as defined by nonstandard models of the first-order theory of the reals. This system of numbers was closely related to the ring of Schmieden and Laugwitz, developed independently a few years earlier.
During the last thirty years the use of nonstandard models in mathematics has taken its rightful place among the various methods employed by mathematicians. The contributions in this volume have been selected to present a panoramic view of the various directions in which nonstandard analysis is advancing, thus serving as a source of inspiration for future research.
Papers have been grouped in sections dealing with analysis, topology and topological groups; probability theory; and mathematical physics.
This volume can be used as a complementary text to courses in nonstandard analysis, and will be of interest to graduate students and researchers in both pure and applied mathematics and physics.https://authors.library.caltech.edu/records/vamh5-1jp67Stable seminorms revisited
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854012
Authors: Arens, Richard; Goldberg, Moshe; Luxemburg, W. A. J.
Year: 1998
DOI: 10.7153/mia-01-02
A seminorm S on an algebra A is called stable if for some constant σ > 0 ,
S(x^k) ≤ σS(x)^k for all x ∈ A and all k = 1, 2, 3,....
We call S strongly stable if the above holds with σ = 1 . In this note we use several known
and new results to shed light on the concepts of stability. In particular, the interrelation between
stability and similar ideas is discussed.https://authors.library.caltech.edu/records/7tndn-k6w87Stable Norms on Complex Numbers and Quaternions
https://resolver.caltech.edu/CaltechAUTHORS:20181003-155853311
Authors: Arens, Richard; Goldberg, Moshe; Luxemburg, W. A. J.
Year: 1999
DOI: 10.1006/jabr.1998.7849
In this paper, we study stability properties of norms on the complex numbers and on the quaternions. Our main findings are that these norms are stable if and only if they majorize the modulus function and that not all stable norms are strongly stable. Part of the paper is devoted to the standard matrix representations of the above number systems, where we show that norms on the corresponding matrix algebras are stable if and only if they are spectrally dominant. We conclude by considering proper seminorms, observing that none are stable on the complex numbers or on the quaternions.https://authors.library.caltech.edu/records/2611n-60496Stable subnorms
https://resolver.caltech.edu/CaltechAUTHORS:20181003-140239218
Authors: Goldberg, Moshe; Luxemburg, W. A. J.
Year: 2000
DOI: 10.1016/s0024-3795(00)00011-2
Let f be a real-valued function defined on a nonempty subset of an algebra over a field , either or , so that is closed under scalar multiplication. Such f shall be called a subnorm on if f(a)>0 for all , and f(αa)=∣α∣f(a) for all and . If in addition, is closed under raising to powers, and f(am)=f(a)m for all and m=1,2,3,…, then f shall be called a submodulus. Further, a subnorm f shall be called stable if there exists a constant σ>0 so that f(am)⩽σf(a)m for all and m=1,2,3,… Our primary purpose in this paper is to study stability properties of continuous subnorms on subsets of finite dimensional algebras. If f is a subnorm on such a set , and g is a continuous submodulus on the same set, then our main results state that g is unique, f(am)1/m→g(a) as m→∞, and f is stable if and only if it majorizes g. In particular, if f is a subnorm on a subset of , the algebra of n×n matrices over , and if has the above properties but no nilpotent elements, then we show that f is stable if and only if it is spectrally dominant, i.e., f(A)⩾ρ(A) for all , where ρ is the spectral radius. Part of the paper is devoted to norms on algebras, where the above findings hold almost verbatim. We illustrate our results by discussing certain subnorms on matrix algebras, as well as on the complex numbers, the quaternions, and the octaves, where these number systems are viewed as algebras over the reals.https://authors.library.caltech.edu/records/4vtv9-96n78Charles Boudewijn Huijsmans 1946–1997
https://resolver.caltech.edu/CaltechAUTHORS:20181003-151847109
Authors: Luxemburg, W. A. J.; de Pagter, B.
Year: 2000
DOI: 10.1023/A:1017242730521
C.B. "Pay" Huijsmans was born in Voorburg (The Netherlands) on May 7, 1946. After finishing the Gymnasium in 1964, Pay entered the University of Leiden majoring in mathematics. In 1970, he started to work on his Ph.D. thesis as a student of Prof. A.C. Zaanen. At that time the Functional Analysis group in Leiden was a center of great activity, particularly in the areas of Banach function spaces, integral operators and the theory of Riesz spaces (vector lattices).https://authors.library.caltech.edu/records/emmp2-s1m26Existence of Non-Trivial Bounded Functionals Implies the Hahn-Banach Extension Theorem
https://resolver.caltech.edu/CaltechAUTHORS:20181003-135833363
Authors: Luxemburg, W. A. J.; Väth, Martin
Year: 2001
DOI: 10.4171/zaa/1015
We show that it is impossible to prove the existence of a linear (bounded or unbounded) functional on any L_∞/C_0 without an uncountable form of the axiom of choice. Moreover, we show that if on each Banach space there exists at least one non-trivial bounded linear functional, then the Hahn-Banach extension theorem must hold. We also discuss relations of non-measurable sets and the Hahn-Banach extension theorem.https://authors.library.caltech.edu/records/yaqh7-k1723Discontinuous subnorms
https://resolver.caltech.edu/CaltechAUTHORS:20181003-135909209
Authors: Goldberg, Moshe; Luxemburg, W. A. J.
Year: 2001
DOI: 10.1080/03081080108818683
Let S be a subset of a finite-dimensional algebra over a field F either R or C so that S is closed under scalar multiplication. A real-valued function f defined on S, shall be called a subnorm if f(a) > 0 for all 0 ≠ a ε S, and f(αa) = for all a ε S and α ε F. If in addition S is closed under raising to powers, and f(am )=f(a)m for all a ε S and m = 1,2,3,⋯, then f shall be called a submodulus. Further, if S is closed under multiplication, then a submodulus f shall be called a modulus if f(ab) = f(a)f(b) for all a,b ε S. Our main purpose in this paper is to construct discontinuous subnorms, submoduli and moduli, on the complex numbers, the quaternions, and on suitable sets of matrices. In each of these cases we discuss the asymptotic behavior and stability properties of the obtained objects.https://authors.library.caltech.edu/records/caqkz-st054Not all quadrative norms are strongly stable
https://resolver.caltech.edu/CaltechAUTHORS:20181003-135909088
Authors: Goldberg, Moshe; Guralnick, Robert; Luxemburg, W. A. J.
Year: 2001
DOI: 10.1016/s0019-3577(01)80035-5
A norm N on an algebra A is called quadrative if N(x^2) ≤ N(x)^2 for all x ∈ A, and strongly stable if N(x^k) ≤ N(x)^k for all x ∈ A and all k = 2, 3, 4…. Our main purpose in this note is to show that not all quadrative norms are strongly stable.https://authors.library.caltech.edu/records/w5ska-9bn44Maharam extensions of positive operators and f-modules
https://resolver.caltech.edu/CaltechAUTHORS:20181003-152735098
Authors: Luxemburg, W. A. J.; de Pagter, B.
Year: 2002
DOI: 10.1023/A:1015249114403
The principal result of this paper is the construction of simultaneous extensions of collections of positive linear operators between vector lattices to interval preserving operators (i.e., Maharam operators). This construction is based on some properties of so-called f-modules. The properties and structure of these extension spaces is discussed in some detail.https://authors.library.caltech.edu/records/be47w-sak74Sundual characterizations of the translation group of R
https://resolver.caltech.edu/CaltechAUTHORS:JACpams03
Authors: Jackson, Frances Y.; Luxemburg, W. A. J.
Year: 2003
DOI: 10.1090/S0002-9939-02-06632-7
We characterize the first three sundual spaces of C-0(R), with respect to the translation group of R.https://authors.library.caltech.edu/records/g87sg-44g83Stable Subnorms II
https://resolver.caltech.edu/CaltechAUTHORS:20181003-135833494
Authors: Goldberg, Moshe; Guralnick, Robert; Luxemburg, W. A. J.
Year: 2003
DOI: 10.1080/0308108031000078920
In this paper we continue our study of stability properties of subnorms on subsets of finite-dimensional, power-associative algebras over the real or the complex numbers.https://authors.library.caltech.edu/records/qw9e2-man37Characterizations of Sunduals (Summary)
https://resolver.caltech.edu/CaltechAUTHORS:20181003-152216402
Authors: Luxemburg, W. A. J.
Year: 2003
DOI: 10.1023/A:1025820100268
[no abstract]https://authors.library.caltech.edu/records/qb29n-f7j21Stable subnorms revisited
https://resolver.caltech.edu/CaltechAUTHORS:GOLpjm04
Authors: Goldberg, Moshe; Luxemburg, W. A. J.
Year: 2004
Let A be a finite-dimensional, power-associative algebra over a field F, either R or C, and let S, a subset of A, be closed under scalar multiplication. A real-valued function f defined on S, shall be called a subnorm if f(a) > 0 for all 0 not equal a is an element of S, and f(alpha a) = |alpha| f(a) for all a is an element of S and alpha is an element of F. If in addition, S is closed under raising to powers, then a subnorm f shall be called stable if there exists a constant sigma > 0 so that f(a(m)) less than or equal to sigma f(a)(m) for all a is an element of S and m = 1, 2, 3....
The purpose of this paper is to provide an updated account of our study of stable subnorms on subsets of finite-dimensional, power-associative algebras over F. Our goal is to review and extend several of our results in two previous papers, dealing mostly with continuous subnorms on closed sets.https://authors.library.caltech.edu/records/rdnb2-2tr45Representations of Positive Projections I
https://resolver.caltech.edu/CaltechAUTHORS:20181003-135738430
Authors: Luxemburg, W. A. J.; de Pagter, B.
Year: 2005
DOI: 10.1007/s11117-004-2773-5
In this paper we start the development of a general theory of Maharam-type representation theorems for positive projections on Dedekind complete vector lattices. In the approach to these results the theory off-algebras plays a crucial role.https://authors.library.caltech.edu/records/711d1-eb515Representations of Positive Projections II
https://resolver.caltech.edu/CaltechAUTHORS:20181003-135738314
Authors: Luxemburg, W. A. J.; de Pagter, B.
Year: 2005
DOI: 10.1007/s11117-004-2774-4
In this paper we obtain a number of Maharam-type slice integral representations, with respect to scalar measures, for positive projections in Dedekind complete vector lattices and f-algebras.https://authors.library.caltech.edu/records/1bem2-aqn27A Distributional proof of a theorem of Plessner
https://resolver.caltech.edu/CaltechAUTHORS:20090910-133620697
Authors: Luxemburg, Wilhelmus A. J.
Year: 2009
DOI: 10.1007/s00013-009-3109-2
The paper presents a proof, using methods of the theory of distributions of the famous result of A. Plessner characterizing the absolutely continuous measures among the class of Borel measures.https://authors.library.caltech.edu/records/ewea5-zkf92Adriaan Cornelis Zaanen
https://resolver.caltech.edu/CaltechAUTHORS:20140326-115616709
Authors: Kaashoek, Marinus; Luxemburg, Wim; de Pagter, Ben
Year: 2014
DOI: 10.1016/j.indag.2013.11.002
Adriaan (Aad) Cornelis Zaanen was born on 14 June 1913 in Rotterdam as the oldest son of Pieter Zaanen and Ariaantje de Bruijn. His father was a building contractor, renovating mainly historical buildings. From 1925 until 1930, Aad attended high school in Rotterdam, where his teacher in mathematics was the brother of the well-known Dutch mathematician J.G. van der Corput. Aad Zaanen was married to Ada Jacoba van der Woude and together they had four sons.
He passed away on 1 April 2003.https://authors.library.caltech.edu/records/mrn51-8a193Notes on the Theory of Integration : Ma 108
https://resolver.caltech.edu/CaltechAUTHORS:20170327-135159965
Authors: Luxemburg, W. A. J.
Year: 2017
Course description from Caltech Catalog (1960/61):
Ma 108 abc. Advanced Calculus. 12 units (4-0-8); three terms. Prerequisite: Ma 2. In this course, a sequel to Ma 2, more advanced techniques and applications of calculus are treated. Point set topology is the point of departure for the theory of convergence,
and applications are made to implicit functions, partial differentiation, infinite series and infinite products of real and complex numbers. Other topics treated include: uniform convergence of sequences of functions; functions defined by integrals; Fourier series and integrals; analytic functions of a complex variable.https://authors.library.caltech.edu/records/k4m2c-khm74Non-Standard Analysis: Lectures on A. Robinson's Theory of Infinitesimals and Infinitely Large Numbers
https://resolver.caltech.edu/CaltechAUTHORS:20170327-135729832
Authors: Luxemburg, W. A. J.
Year: 2017
The present lecture notes have grown from a series of three lectures which were given by the author at the California Institute of Technology in December 1961. The purpose of these lectures was to give a discussion of A. Robinson's theory of infinitesimals and infinitely large numbers which had just appeared in print under the title "Non-Standard Analysis".
The title "Non-Standard Analysis" refers to the fact that this theory is an interpretation of analysis in a non-standard model of the arithmetic of the real numbers.https://authors.library.caltech.edu/records/pp04a-k3d74Robinson, 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