[
{
"id": "authors:n9wt3-s4178",
"collection": "authors",
"collection_id": "n9wt3-s4178",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210203-081026434",
"type": "book_section",
"title": "Robinson, Abraham (1918\u20131974)",
"book_title": "The New Palgrave Dictionary of Economics",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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.",
"doi": "10.1057/978-1-349-95189-5_1579",
"isbn": "978-1-349-95189-5",
"publisher": "Palgrave Macmillan UK",
"place_of_publication": "London",
"publication_date": "2018-02-15",
"pages": "11763-11764"
},
{
"id": "authors:pp04a-k3d74",
"collection": "authors",
"collection_id": "pp04a-k3d74",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170327-135729832",
"type": "teaching_resource",
"title": "Non-Standard Analysis: Lectures on A. Robinson's Theory of Infinitesimals and Infinitely Large Numbers",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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\".\nThe 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.",
"publisher": "Mathematics Department. California Institute of Technology",
"publication_date": "2017-03-27"
},
{
"id": "authors:k4m2c-khm74",
"collection": "authors",
"collection_id": "k4m2c-khm74",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170327-135159965",
"type": "teaching_resource",
"title": "Notes on the Theory of Integration : Ma 108",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "Course description from Caltech Catalog (1960/61):\n\nMa 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,\nand 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.",
"publisher": "Mathematics Department. California Institute of Technology",
"publication_date": "2017-03-27"
},
{
"id": "authors:mrn51-8a193",
"collection": "authors",
"collection_id": "mrn51-8a193",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140326-115616709",
"type": "article",
"title": "Adriaan Cornelis Zaanen",
"author": [
{
"family_name": "Kaashoek",
"given_name": "Marinus",
"clpid": "Kaashoek-M"
},
{
"family_name": "Luxemburg",
"given_name": "Wim",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "de Pagter",
"given_name": "Ben",
"clpid": "de-Pagter-B"
}
],
"abstract": "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.\nHe passed away on 1 April 2003.",
"doi": "10.1016/j.indag.2013.11.002",
"issn": "0019-3577",
"publisher": "Elsevier",
"publication": "Indagationes Mathematicae",
"publication_date": "2014-03-14",
"series_number": "2",
"volume": "25",
"issue": "2",
"pages": "164-169"
},
{
"id": "authors:ewea5-zkf92",
"collection": "authors",
"collection_id": "ewea5-zkf92",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20090910-133620697",
"type": "article",
"title": "A Distributional proof of a theorem of Plessner",
"author": [
{
"family_name": "Luxemburg",
"given_name": "Wilhelmus A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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.",
"doi": "10.1007/s00013-009-3109-2",
"issn": "0003-889X",
"publisher": "Springer",
"publication": "Archiv der Mathematik",
"publication_date": "2009-05",
"series_number": "5",
"volume": "92",
"issue": "5",
"pages": "501-503"
},
{
"id": "authors:1bem2-aqn27",
"collection": "authors",
"collection_id": "1bem2-aqn27",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-135738314",
"type": "article",
"title": "Representations of Positive Projections II",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "de Pagter",
"given_name": "B.",
"clpid": "de-Pagter-B"
}
],
"abstract": "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.",
"doi": "10.1007/s11117-004-2774-4",
"issn": "1385-1292",
"publisher": "Springer",
"publication": "Positivity",
"publication_date": "2005-12",
"series_number": "4",
"volume": "9",
"issue": "4",
"pages": "569-605"
},
{
"id": "authors:711d1-eb515",
"collection": "authors",
"collection_id": "711d1-eb515",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-135738430",
"type": "article",
"title": "Representations of Positive Projections I",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "de Pagter",
"given_name": "B.",
"clpid": "de-Pagter-B"
}
],
"abstract": "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.",
"doi": "10.1007/s11117-004-2773-5",
"issn": "1385-1292",
"publisher": "Springer",
"publication": "Positivity",
"publication_date": "2005-09",
"series_number": "3",
"volume": "9",
"issue": "3",
"pages": "293-325"
},
{
"id": "authors:rdnb2-2tr45",
"collection": "authors",
"collection_id": "rdnb2-2tr45",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:GOLpjm04",
"type": "article",
"title": "Stable subnorms revisited",
"author": [
{
"family_name": "Goldberg",
"given_name": "Moshe",
"clpid": "Goldberg-M"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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....\n\nThe 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.",
"issn": "0030-8730",
"publisher": "Pacific Journal of Mathematics",
"publication": "Pacific Journal of Mathematics",
"publication_date": "2004-05-01",
"series_number": "1",
"volume": "215",
"issue": "1",
"pages": "15-27"
},
{
"id": "authors:qb29n-f7j21",
"collection": "authors",
"collection_id": "qb29n-f7j21",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-152216402",
"type": "article",
"title": "Characterizations of Sunduals (Summary)",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "[no abstract]",
"doi": "10.1023/A:1025820100268",
"issn": "1385-1292",
"publisher": "Springer",
"publication": "Positivity",
"publication_date": "2003-06",
"series_number": "1-2",
"volume": "7",
"issue": "1-2",
"pages": "81-85"
},
{
"id": "authors:qw9e2-man37",
"collection": "authors",
"collection_id": "qw9e2-man37",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-135833494",
"type": "article",
"title": "Stable Subnorms II",
"author": [
{
"family_name": "Goldberg",
"given_name": "Moshe",
"clpid": "Goldberg-M"
},
{
"family_name": "Guralnick",
"given_name": "Robert",
"clpid": "Guralnick-R"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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.",
"doi": "10.1080/0308108031000078920",
"issn": "0308-1087",
"publisher": "Informa UK Limited",
"publication": "Linear and Multilinear Algebra",
"publication_date": "2003-06",
"series_number": "2",
"volume": "51",
"issue": "2",
"pages": "209-219"
},
{
"id": "authors:g87sg-44g83",
"collection": "authors",
"collection_id": "g87sg-44g83",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:JACpams03",
"type": "article",
"title": "Sundual characterizations of the translation group of R",
"author": [
{
"family_name": "Jackson",
"given_name": "Frances Y.",
"clpid": "Jackson-F-Y"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "We characterize the first three sundual spaces of C-0(R), with respect to the translation group of R.",
"doi": "10.1090/S0002-9939-02-06632-7",
"issn": "0002-9939",
"publisher": "American Mathematical Society",
"publication": "Proceedings of the American Mathematical Society",
"publication_date": "2003-01",
"series_number": "1",
"volume": "131",
"issue": "1",
"pages": "185-199"
},
{
"id": "authors:be47w-sak74",
"collection": "authors",
"collection_id": "be47w-sak74",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-152735098",
"type": "article",
"title": "Maharam extensions of positive operators and f-modules",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "de Pagter",
"given_name": "B.",
"clpid": "de-Pagter-B"
}
],
"abstract": "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.",
"doi": "10.1023/A:1015249114403",
"issn": "1385-1292",
"publisher": "Springer",
"publication": "Positivity",
"publication_date": "2002-06",
"series_number": "2",
"volume": "6",
"issue": "2",
"pages": "147-190"
},
{
"id": "authors:w5ska-9bn44",
"collection": "authors",
"collection_id": "w5ska-9bn44",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-135909088",
"type": "article",
"title": "Not all quadrative norms are strongly stable",
"author": [
{
"family_name": "Goldberg",
"given_name": "Moshe",
"clpid": "Goldberg-M"
},
{
"family_name": "Guralnick",
"given_name": "Robert",
"clpid": "Guralnick-R"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "A norm N on an algebra A is called quadrative if N(x^2) \u2264 N(x)^2 for all x \u2208 A, and strongly stable if N(x^k) \u2264 N(x)^k for all x \u2208 A and all k = 2, 3, 4\u2026. Our main purpose in this note is to show that not all quadrative norms are strongly stable.",
"doi": "10.1016/s0019-3577(01)80035-5",
"issn": "0019-3577",
"publisher": "Elsevier",
"publication": "Indagationes Mathematicae",
"publication_date": "2001-12-17",
"series_number": "4",
"volume": "12",
"issue": "4",
"pages": "469-476"
},
{
"id": "authors:yaqh7-k1723",
"collection": "authors",
"collection_id": "yaqh7-k1723",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-135833363",
"type": "article",
"title": "Existence of Non-Trivial Bounded Functionals Implies the Hahn-Banach Extension Theorem",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "V\u00e4th",
"given_name": "Martin",
"clpid": "V\u00e4th-M"
}
],
"abstract": "We show that it is impossible to prove the existence of a linear (bounded or unbounded) functional on any L_\u221e/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.",
"doi": "10.4171/zaa/1015",
"issn": "0232-2064",
"publisher": "European Mathematical Publishing House",
"publication": "Zeitschrift f\u00fcr Analysis und ihre Anwendungen",
"publication_date": "2001",
"series_number": "2",
"volume": "20",
"issue": "2",
"pages": "267-279"
},
{
"id": "authors:caqkz-st054",
"collection": "authors",
"collection_id": "caqkz-st054",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-135909209",
"type": "article",
"title": "Discontinuous subnorms",
"author": [
{
"family_name": "Goldberg",
"given_name": "Moshe",
"clpid": "Goldberg-M"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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 \u2260 a \u03b5 S, and f(\u03b1a) = for all a \u03b5 S and \u03b1 \u03b5 F. If in addition S is closed under raising to powers, and f(am )=f(a)m for all a \u03b5 S and m = 1,2,3,\u22ef, 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 \u03b5 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.",
"doi": "10.1080/03081080108818683",
"issn": "0308-1087",
"publisher": "Informa UK Limited",
"publication": "Linear and Multilinear Algebra",
"publication_date": "2001",
"series_number": "1",
"volume": "49",
"issue": "1",
"pages": "1-24"
},
{
"id": "authors:emmp2-s1m26",
"collection": "authors",
"collection_id": "emmp2-s1m26",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-151847109",
"type": "article",
"title": "Charles Boudewijn Huijsmans 1946\u20131997",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "de Pagter",
"given_name": "B.",
"clpid": "de-Pagter-Ben"
}
],
"abstract": "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).",
"doi": "10.1023/A:1017242730521",
"issn": "1385-1292",
"publisher": "Springer",
"publication": "Positivity",
"publication_date": "2000-09",
"series_number": "3",
"volume": "4",
"issue": "3",
"pages": "203-204"
},
{
"id": "authors:4vtv9-96n78",
"collection": "authors",
"collection_id": "4vtv9-96n78",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-140239218",
"type": "article",
"title": "Stable subnorms",
"author": [
{
"family_name": "Goldberg",
"given_name": "Moshe",
"clpid": "Goldberg-M"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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(\u03b1a)=\u2223\u03b1\u2223f(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,\u2026, then f shall be called a submodulus. Further, a subnorm f shall be called stable if there exists a constant \u03c3>0 so that f(am)\u2a7d\u03c3f(a)m for all and m=1,2,3,\u2026 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\u2192g(a) as m\u2192\u221e, 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\u00d7n 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)\u2a7e\u03c1(A) for all , where \u03c1 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.",
"doi": "10.1016/s0024-3795(00)00011-2",
"issn": "0024-3795",
"publisher": "Elsevier",
"publication": "Linear Algebra and its Applications",
"publication_date": "2000-03-01",
"series_number": "1-3",
"volume": "307",
"issue": "1-3",
"pages": "89-101"
},
{
"id": "authors:2611n-60496",
"collection": "authors",
"collection_id": "2611n-60496",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155853311",
"type": "article",
"title": "Stable Norms on Complex Numbers and Quaternions",
"author": [
{
"family_name": "Arens",
"given_name": "Richard",
"clpid": "Arens-R"
},
{
"family_name": "Goldberg",
"given_name": "Moshe",
"clpid": "Goldberg-M"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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.",
"doi": "10.1006/jabr.1998.7849",
"issn": "0021-8693",
"publisher": "Elsevier",
"publication": "Journal of Algebra",
"publication_date": "1999-09-01",
"series_number": "1",
"volume": "219",
"issue": "1",
"pages": "1-15"
},
{
"id": "authors:7tndn-k6w87",
"collection": "authors",
"collection_id": "7tndn-k6w87",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854012",
"type": "article",
"title": "Stable seminorms revisited",
"author": [
{
"family_name": "Arens",
"given_name": "Richard",
"clpid": "Arens-R"
},
{
"family_name": "Goldberg",
"given_name": "Moshe",
"clpid": "Goldberg-M"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "A seminorm S on an algebra A is called stable if for some constant \u03c3 > 0 ,\nS(x^k) \u2264 \u03c3S(x)^k for all x \u2208 A and all k = 1, 2, 3,....\nWe call S strongly stable if the above holds with \u03c3 = 1 . In this note we use several known\nand new results to shed light on the concepts of stability. In particular, the interrelation between\nstability and similar ideas is discussed.",
"doi": "10.7153/mia-01-02",
"issn": "1331-4343",
"publisher": "Element d.o.o.",
"publication": "Mathematical Inequalities & Applications",
"publication_date": "1998",
"series_number": "1",
"volume": "1",
"issue": "1",
"pages": "31-40"
},
{
"id": "authors:fqayp-zj228",
"collection": "authors",
"collection_id": "fqayp-zj228",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155855535",
"type": "book_section",
"title": "Diagonals of the Powers of an Operator on a Banach Lattice",
"book_title": "Operator Theory in Function Spaces and Banach Lattices",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "de Pagter",
"given_name": "B.",
"clpid": "de-Pagter-B"
},
{
"family_name": "Schep",
"given_name": "A. R.",
"clpid": "Schep-A-R"
}
],
"contributor": [
{
"family_name": "Huijsmans",
"given_name": "C. B.",
"clpid": "Huijsmans-C-B"
},
{
"family_name": "Kaashoek",
"given_name": "M. A.",
"clpid": "Kaashoek-M-A"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "de Pagter",
"given_name": "B.",
"clpid": "de-Pagter-B"
}
],
"abstract": "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)\noperators 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| \u2264 \u03bbI 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.",
"doi": "10.1007/978-3-0348-9076-2_13",
"isbn": "9783034898966",
"publisher": "Birkh\u00e4user Basel",
"place_of_publication": "Basel",
"publication_date": "1995",
"pages": "223-273"
},
{
"id": "authors:7g4qz-srh26",
"collection": "authors",
"collection_id": "7g4qz-srh26",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854110",
"type": "book_section",
"title": "Biographical Notes",
"book_title": "Operator Theory in Function Spaces and Banach Lattices",
"author": [
{
"family_name": "Huijsmans",
"given_name": "C. B.",
"clpid": "Huijsmans-C-B"
},
{
"family_name": "Kaashoek",
"given_name": "M. A.",
"clpid": "Kaashoek-M-A"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "de Pagter",
"given_name": "B.",
"clpid": "de-Pagter-B"
}
],
"contributor": [
{
"family_name": "Huijsmans",
"given_name": "C. B.",
"clpid": "Huijsmans-C-B"
},
{
"family_name": "Kaashoek",
"given_name": "M. A.",
"clpid": "Kaashoek-M-A"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "de Pagter",
"given_name": "B.",
"clpid": "de-Pagter-B"
}
],
"abstract": "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.",
"doi": "10.1007/978-3-0348-9076-2_1",
"isbn": "9783034898966",
"publisher": "Birkh\u00e4user Basel",
"place_of_publication": "Basel",
"publication_date": "1995",
"pages": "1-5"
},
{
"id": "authors:vamh5-1jp67",
"collection": "authors",
"collection_id": "vamh5-1jp67",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181012-074421607",
"type": "book",
"title": "Advances in Analysis, Probability and Mathematical Physics: Contributions of Nonstandard Analysis",
"author": [
{
"family_name": "Albeverio",
"given_name": "Sergio A."
},
{
"family_name": "Luxemburg",
"given_name": "Wilhelm A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "Wolff",
"given_name": "Manfred P. H."
}
],
"contributor": [
{
"family_name": "Albeverio",
"given_name": "Sergio A.",
"clpid": "Albeverio-S-A"
},
{
"family_name": "Luxemburg",
"given_name": "Wilhelm A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "Wolff",
"given_name": "Manfred P. H.",
"clpid": "Wolff-M-P-H"
}
],
"abstract": "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. \nDuring 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. \nPapers have been grouped in sections dealing with analysis, topology and topological groups; probability theory; and mathematical physics. \nThis 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.",
"doi": "10.1007/978-94-015-8451-7",
"isbn": "978-90-481-4481-5",
"publisher": "Springer",
"place_of_publication": "Dordrecht",
"publication_date": "1995"
},
{
"id": "authors:yn8wc-nq866",
"collection": "authors",
"collection_id": "yn8wc-nq866",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181011-162816470",
"type": "book",
"title": "Operator Theory in Function Spaces and Banach Lattices: Essays dedicated to A.C. Zaanen on the occasion of his 80th birthday",
"author": [
{
"family_name": "Huijsmans",
"given_name": "C. B."
},
{
"family_name": "Kaashoek",
"given_name": "M. A."
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "de Pagter",
"given_name": "B."
}
],
"contributor": [
{
"family_name": "Huijsmans",
"given_name": "C. B.",
"clpid": "Huijsmans-C-B"
},
{
"family_name": "Kaashoek",
"given_name": "M. A.",
"clpid": "Kaashoek-M-A"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "de Pagter",
"given_name": "B.",
"clpid": "de-Pagter-B"
}
],
"abstract": "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.",
"doi": "10.1007/978-3-0348-9076-2",
"isbn": "978-3-0348-9896-6",
"publisher": "Springer",
"place_of_publication": "Basel, Switzerland",
"publication_date": "1995"
},
{
"id": "authors:dcnbx-9tc44",
"collection": "authors",
"collection_id": "dcnbx-9tc44",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-131120934",
"type": "article",
"title": "Multiplicativity Factors for Orlicz Space Function Norms",
"author": [
{
"family_name": "Arens",
"given_name": "Richard",
"clpid": "Arens-R"
},
{
"family_name": "Goldberg",
"given_name": "Moshe",
"clpid": "Goldberg-M"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "Let \u03c1\u03c6 be a function norm defined by a Young function \u03c6 with respect to a measure space (T, \u03a9, m), and let L\u03c6 be the Orlicz space determined by \u03c1\u03c6. If L\u03c6 is an algebra, then a constant \u03bc > 0 is called a multiplicativity factor for \u03c1\u03c6, if \u03c1\u03c6,(fg) \u2264 \u03bc\u03c1\u03c6(f) \u03c1\u03c6(g) for all f, g \u2208 L\u03c6. The main objective of this paper is to give conditions under which L\u03c6 is indeed an algebra, and to obtain in this case the best (least) multiplicativity factor for \u03c1\u03c6. The first of our principal results is that L\u03c6 is an algebra if and only if or Our second main result states that if L\u03c6 is an algebra and (T, \u03a9, m) is free of infinite atoms, then the best multiplicativity factor for \u03c1\u03c6 is \u03c6\u22121(1/minf if minf > 0, and x\u221e(\u03c6) if minf = 0.",
"doi": "10.1006/jmaa.1993.1264",
"issn": "0022-247X",
"publisher": "Elsevier",
"publication": "Journal of Mathematical Analysis and Applications",
"publication_date": "1993-08",
"series_number": "2",
"volume": "177",
"issue": "2",
"pages": "386-411"
},
{
"id": "authors:60ptd-2r168",
"collection": "authors",
"collection_id": "60ptd-2r168",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-131548532",
"type": "article",
"title": "Multiplicativity Factors for Function Norms",
"author": [
{
"family_name": "Arens",
"given_name": "R.",
"clpid": "Arens-R"
},
{
"family_name": "Goldberg",
"given_name": "M.",
"clpid": "Goldberg-M"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "Let (T, \u03a9, m) be a measure space; let \u03c1 be a function norm on = (T, \u03a9, m), the algebra of measurable functions on T; and let L\u03c1 be the space {f \u2208 : \u03c1(f) < \u221e} modulo the null functions. If L\u03c1, is an algebra, then we call a constant \u03bc > 0 a multiplicativity factor for \u03c1 if \u03c1(fg) \u2264 \u03bc\u03c1(f) \u03c1(g) for all f, g \u2208 L\u03c1. Similarly, \u03bb > 0 is a quadrativity factor if \u03c1(f2) \u2264 \u03bb\u03c1(f)2 for all f. The main purpose of this paper is to give conditions under which L\u03c1, is indeed an algebra, and to obtain in this case the best (least) multiplicativity and quadrativity factors for \u03c1. The first of our two principal results is that if \u03c1 is \u03c3-subadditive, then L\u03c1 is an algebra if and only if L\u03c1 is contained in L\u221e. Our second main result is that if (T, \u03a9, m) is free of infinite atoms, \u03c1 is \u03c3-subadditive and saturated, and L\u03c1, is an algebra, then the multiplicativity and quadrativity factors for \u03c1 coincide, and the best such factor is determined by sup{||f||\u221e: f \u2208 L\u03c1, \u03c1(f) \u2264 1}.",
"doi": "10.1006/jmaa.1993.1263",
"issn": "0022-247X",
"publisher": "Elsevier",
"publication": "Journal of Mathematical Analysis and Applictions",
"publication_date": "1993-08",
"series_number": "2",
"volume": "177",
"issue": "2",
"pages": "368-385"
},
{
"id": "authors:69hyp-jeq94",
"collection": "authors",
"collection_id": "69hyp-jeq94",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-132007385",
"type": "article",
"title": "Multiplicativity factors for seminorms. II",
"author": [
{
"family_name": "Arens",
"given_name": "Richard",
"clpid": "Arens-R"
},
{
"family_name": "Goldberg",
"given_name": "Moshe",
"clpid": "Goldberg-M"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "Let S be a seminorm on an algebra . In this paper we study multiplicativity and quadrativity factors for S, i.e., constants \u03bc > 0 and \u03bb > 0 for which S(xy) \u2a7d \u03bcS(x)S(y) and S(x2) \u2a7d \u03bbS(x)2 for all x, y \u2208 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.",
"doi": "10.1016/0022-247X(92)90026-A",
"issn": "0022-247X",
"publisher": "Elsevier",
"publication": "Journal of Mathematical Analysis and Applictions",
"publication_date": "1992-11",
"series_number": "2",
"volume": "170",
"issue": "2",
"pages": "401-413"
},
{
"id": "authors:82syf-ajk89",
"collection": "authors",
"collection_id": "82syf-ajk89",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155855350",
"type": "article",
"title": "An Alternative Proof of a Radon-Nikodym Theorem for Lattice Homomorphisms",
"author": [
{
"family_name": "Huijsmans",
"given_name": "C. B.",
"clpid": "Huijsmans-C-B"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "We give a new proof of the Luxemburg-Schep theorem for lattice homomorphisms.",
"doi": "10.1007/978-94-017-2721-1_7",
"issn": "0167-8019",
"isbn": "9789048142057",
"publisher": "Springer",
"place_of_publication": "Dordrecht",
"publication": "Acta Applicandae Mathematicae",
"publication_date": "1992-05",
"series_number": "1-2",
"volume": "27",
"issue": "1-2",
"pages": "67-71"
},
{
"id": "authors:tpkdd-f1132",
"collection": "authors",
"collection_id": "tpkdd-f1132",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200515-115519213",
"type": "book_section",
"title": "An Alternative Proof of a Radon-Nikodym Theorem for Lattice Homomorphisms",
"book_title": "Positive Operators and Semigroups on Banach Lattices",
"author": [
{
"family_name": "Huijsmans",
"given_name": "C. B.",
"clpid": "Huijsmans-C-B"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"contributor": [
{
"family_name": "Huijsmans",
"given_name": "C. B.",
"clpid": "Huijsmans-C-B"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "We give a new proof of the Luxemburg-Schep theorem for lattice homomorphisms.",
"doi": "10.1007/978-94-017-2721-1_7",
"isbn": "9789048142057",
"publisher": "Springer Netherlands",
"place_of_publication": "Dordrecht",
"publication_date": "1992",
"pages": "67-71"
},
{
"id": "authors:gnkjs-phb84",
"collection": "authors",
"collection_id": "gnkjs-phb84",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181011-162006203",
"type": "book",
"title": "Positive Operators and Semigroups on Banach Lattices: Proceedings of a Caribbean Mathematics Foundation Conference 1990",
"author": [
{
"family_name": "Huijsmans",
"given_name": "C. B."
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"contributor": [
{
"family_name": "Huijsmans",
"given_name": "C. B.",
"clpid": "Huijsmans-C-B"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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. \nThis book will be of interest to analysts whose work involves positive matrices and positive operators.",
"doi": "10.1007/978-94-017-2721-1",
"isbn": "978-90-481-4205-7",
"publisher": "Springer",
"place_of_publication": "Dordrecht",
"publication_date": "1992"
},
{
"id": "authors:wcnfe-2bg85",
"collection": "authors",
"collection_id": "wcnfe-2bg85",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200515-130605911",
"type": "book",
"title": "Positive Operators, Riesz Spaces, and Economics",
"book_title": "Positive Operators, Riesz Spaces, and Economics",
"author": [
{
"family_name": "Aliprantis",
"given_name": "Charalambos D.",
"clpid": "Aliprantis-C-D"
},
{
"family_name": "Border",
"given_name": "Kim C.",
"orcid": "0000-0003-4437-0524",
"clpid": "Border-K-C"
},
{
"family_name": "Luxemburg",
"given_name": "Wilhelmus A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"contributor": [
{
"family_name": "Aliprantis",
"given_name": "Charalambos D.",
"clpid": "Aliprantis-C-D"
},
{
"family_name": "Border",
"given_name": "Kim C.",
"clpid": "Border-K-C"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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\u00ad tors, and their Applications to Economics took place at the California Institute of Technology. The purpose of the conference was to bring mathematicians special\u00ad ized in Riesz Spaces and economists specialized in General Equilibrium together to exchange ideas and advance the interdisciplinary cooperation between math\u00ad 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\u00ad 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\u00ad ence.",
"doi": "10.1007/978-3-642-58199-1",
"isbn": "978-3-642-63502-1",
"publisher": "Springer",
"place_of_publication": "Berlin",
"publication_date": "1991"
},
{
"id": "authors:bmdz0-mrg72",
"collection": "authors",
"collection_id": "bmdz0-mrg72",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155855248",
"type": "book_section",
"title": "Integration with Respect to Finitely Additive Measures",
"book_title": "Positive Operators, Riesz Spaces, and Economics: Proceedings of a Conference at Caltech, Pasadena, California, April 16\u201320, 1990",
"author": [
{
"family_name": "Luxemburg",
"given_name": "Wilhelmus A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"contributor": [
{
"family_name": "Aliprantis",
"given_name": "Charalambos D.",
"clpid": "Aliprantis-C-D"
},
{
"family_name": "Border",
"given_name": "Kim C.",
"clpid": "Border-K-C"
},
{
"family_name": "Luxemburg",
"given_name": "Wilhelmus A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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. \n\nThe nature of the classical axiom of countable additivity is examined from Carath\u00e9odory's algebraic measure-theoretic point of view.",
"doi": "10.1007/978-3-642-58199-1_6",
"isbn": "9783642635021",
"publisher": "Springer",
"place_of_publication": "Berlin",
"publication_date": "1991",
"pages": "109-150"
},
{
"id": "authors:p6hyp-bgv06",
"collection": "authors",
"collection_id": "p6hyp-bgv06",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181011-142851140",
"type": "book",
"title": "Positive Operators, Riesz Spaces, and Economics",
"author": [
{
"family_name": "Aliprantis",
"given_name": "Charalambos D.",
"clpid": "Aliprantis-C-D"
},
{
"family_name": "Border",
"given_name": "Kim C.",
"orcid": "0000-0003-4437-0524",
"clpid": "Border-K-C"
},
{
"family_name": "Luxemburg",
"given_name": "Wilhelmus A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"contributor": [
{
"family_name": "Aliprantis",
"given_name": "Charalambos D.",
"clpid": "Aliprantis-C-D"
},
{
"family_name": "Border",
"given_name": "Kim C.",
"clpid": "Border-K-C"
},
{
"family_name": "Luxemburg",
"given_name": "Wilhelmus A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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\u00ad tors, and their Applications to Economics took place at the California Institute of Technology. The purpose of the conference was to bring mathematicians special\u00ad ized in Riesz Spaces and economists specialized in General Equilibrium together to exchange ideas and advance the interdisciplinary cooperation between math\u00ad 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\u00ad 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.",
"doi": "10.1007/978-3-642-58199-1",
"isbn": "978-3-642-63502-1",
"publisher": "Springer",
"place_of_publication": "Berlin",
"publication_date": "1991"
},
{
"id": "authors:349kw-39839",
"collection": "authors",
"collection_id": "349kw-39839",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155855154",
"type": "book_section",
"title": "Robinson's Nonstandard Analysis and its Influence on Mathematics",
"book_title": "Rechnen mit dem Unendlichen: Beitr\u00e4ge zur Entwicklung eines kontroversen Gegenstandes",
"author": [
{
"family_name": "Luxemburg",
"given_name": "Wilhelmus A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"contributor": [
{
"family_name": "Spalt",
"given_name": "Detlef D.",
"clpid": "Spalt-D-D"
}
],
"abstract": "The theme of the Tagung concerns the question: Resultatismus oder Geneseologie? Impulse der Nichtstandard Analysis f\u00fcr die Geschichtsforschung.",
"doi": "10.1007/978-3-0348-5242-5_2",
"isbn": "9783034852432",
"publisher": "Birkh\u00e4user",
"place_of_publication": "Basel",
"publication_date": "1990",
"pages": "13-21"
},
{
"id": "authors:g19p0-n7c52",
"collection": "authors",
"collection_id": "g19p0-n7c52",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-132515650",
"type": "article",
"title": "The Work of Dorothy Maharam on Kernel Representations of Linear Operators",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "[no abstract]",
"doi": "10.1090/conm/094/1012988",
"issn": "0271-4132",
"publisher": "American Mathematical Society",
"publication": "Contemporary Mathematics",
"publication_date": "1989",
"volume": "94",
"pages": "177-183"
},
{
"id": "authors:65e8h-mfr78",
"collection": "authors",
"collection_id": "65e8h-mfr78",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854487",
"type": "book_section",
"title": "Robinson, Abraham (1918\u20131974)",
"book_title": "The New Palgrave Dictionary of Economics",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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.",
"doi": "10.1057/978-1-349-95121-5_1579-1",
"isbn": "9781349951215",
"publisher": "Palgrave Macmillan UK",
"place_of_publication": "London",
"publication_date": "1987",
"pages": "1-2"
},
{
"id": "authors:rqbsj-s9010",
"collection": "authors",
"collection_id": "rqbsj-s9010",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200519-134358780",
"type": "book_section",
"title": "Non-Standard Analysis",
"book_title": "Logic, Foundations of Mathematics, and Computability Theory",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"contributor": [
{
"family_name": "Butts",
"given_name": "Robert E.",
"clpid": "Butts-R-E"
},
{
"family_name": "Hintikka",
"given_name": "Jaakko",
"clpid": "Hintikka-J"
}
],
"abstract": "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 \u03f5, \u03b4-method of Weierstrass.",
"doi": "10.1007/978-94-010-1138-9_6",
"isbn": "978-94-010-1140-2",
"publisher": "Springer",
"place_of_publication": "Dordrecht",
"publication_date": "1977",
"pages": "107-119"
},
{
"id": "authors:e66gg-9q404",
"collection": "authors",
"collection_id": "e66gg-9q404",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854296",
"type": "article",
"title": "On a class of valuation fields introduced by A. Robinson",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "It is shown that the nonarchimedean valuation fields ^\u03c1R 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 ^\u03c1E over ^\u03c1R in the sense of Monna which is spherically complete and extends E.",
"doi": "10.1007/bf02756999",
"issn": "0021-2172",
"publisher": "Springer",
"publication": "Israel Journal of Mathematics",
"publication_date": "1976-09",
"series_number": "3-4",
"volume": "25",
"issue": "3-4",
"pages": "189-201"
},
{
"id": "authors:4vzvm-e4c14",
"collection": "authors",
"collection_id": "4vzvm-e4c14",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-135412183",
"type": "article",
"title": "On an infinite series of Abel occurring in the theory of interpolation",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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 (\u2212\u221e, \u22121], the Abel series ! is convergent for all \u03b2>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 \u03b2>0 approximates f uniformly in half-planes of the form Re(z) \u2a7e \u2212 1 + \u03b4, \u03b4>0. At the end of the paper some special cases are discussed.",
"doi": "10.1016/0021-9045(75)90020-9",
"issn": "0021-9045",
"publisher": "Elsevier",
"publication": "Journal of Approximation Theory",
"publication_date": "1975-04",
"series_number": "4",
"volume": "13",
"issue": "4",
"pages": "363-374"
},
{
"id": "authors:p4w8d-p7k62",
"collection": "authors",
"collection_id": "p4w8d-p7k62",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155855059",
"type": "book_section",
"title": "Closure properties of sequences of exponentals { exp (i \u03bb_n t) }",
"book_title": "Topics in Analysis: Colloquium on Mathematical Analysis Jyv\u00e4skyl\u00e4 1970",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"contributor": [
{
"family_name": "Lehto",
"given_name": "Olli",
"clpid": "Lehto-O"
},
{
"family_name": "Louhivaara",
"given_name": "Ilppo Simo",
"clpid": "Louhivaara-I-S"
},
{
"family_name": "Nevanlinna",
"given_name": "Rolf",
"clpid": "Nevanlinna-R"
}
],
"abstract": "[no abstract]",
"doi": "10.1007/bfb0064735",
"isbn": "9783540069652",
"publisher": "Springer",
"place_of_publication": "Berlin",
"publication_date": "1974",
"pages": "268-283"
},
{
"id": "authors:gxhx2-70092",
"collection": "authors",
"collection_id": "gxhx2-70092",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-135810847",
"type": "article",
"title": "What is nonstandard analysis?",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "[no abstract]",
"doi": "10.2307/3038221",
"issn": "0002-9890",
"publisher": "Mathematical Association of America",
"publication": "American Mathematical Monthly",
"publication_date": "1973-06",
"series_number": "6",
"volume": "80",
"issue": "6",
"pages": "38-67"
},
{
"id": "authors:02c8y-pek72",
"collection": "authors",
"collection_id": "02c8y-pek72",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-140410509",
"type": "article",
"title": "On an inequality of A. Khintchine for zero-one matrices",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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,\u2026, m) and column j of sum sj (j = 1, 2,\u2026, n). Then a result of Khintchine states that , where l = max(m, n) and \u03c3 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.",
"doi": "10.1016/0097-3165(72)90043-X",
"issn": "0097-3165",
"publisher": "Elsevier",
"publication": "Journal of Combinatorial Theory. Series A",
"publication_date": "1972-03",
"series_number": "2",
"volume": "12",
"issue": "2",
"pages": "289-296"
},
{
"id": "authors:kvwyq-waq50",
"collection": "authors",
"collection_id": "kvwyq-waq50",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-140756074",
"type": "article",
"title": "Arzela's Dominated Convergence Theorem for the Riemann Integral",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "[no abstract]",
"doi": "10.2307/2317801",
"issn": "0002-9890",
"publisher": "Mathematical Association of America",
"publication": "American Mathematical Monthly",
"publication_date": "1971-11",
"series_number": "9",
"volume": "78",
"issue": "9",
"pages": "970-979"
},
{
"id": "authors:k8dpn-thx56",
"collection": "authors",
"collection_id": "k8dpn-thx56",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-141126373",
"type": "article",
"title": "Entire Functions and Muntz-Szasz Type Approximation",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "Korevaar",
"given_name": "J.",
"clpid": "Korevaar-J"
}
],
"abstract": "Let [a, b] be a bounded interval with a>O. Under what conditions on\n 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.",
"doi": "10.2307/1995828",
"issn": "0002-9947",
"publisher": "American Mathematical Society",
"publication": "Transactions of the American Mathematical Society",
"publication_date": "1971-06",
"series_number": "6",
"volume": "157",
"issue": "6",
"pages": "23-37"
},
{
"id": "authors:jyrkc-9sf89",
"collection": "authors",
"collection_id": "jyrkc-9sf89",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854959",
"type": "article",
"title": "On some order properties of Riesz spaces and their relations",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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.",
"doi": "10.1007/bf01898770",
"issn": "0003-889X",
"publisher": "Springer",
"publication": "Archiv der Mathematik",
"publication_date": "1968-12",
"series_number": "5",
"volume": "19",
"issue": "5",
"pages": "488-493"
},
{
"id": "authors:133nf-08b65",
"collection": "authors",
"collection_id": "133nf-08b65",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-143334534",
"type": "article",
"title": "Is every integral normal?",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "[no abstract]",
"doi": "10.1090/S0002-9904-1967-11825-1",
"issn": "0273-0979",
"publisher": "American Mathematical Society",
"publication": "Bulletin of the American Mathematical Society",
"publication_date": "1967-05",
"series_number": "5",
"volume": "73",
"issue": "5",
"pages": "685-688"
},
{
"id": "authors:e8svt-2q535",
"collection": "authors",
"collection_id": "e8svt-2q535",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-142824090",
"type": "article",
"title": "An extension of the concept of the order dual of a Riesz space",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "Masterson",
"given_name": "J. J.",
"clpid": "Masterson-J-J"
}
],
"abstract": "[no abstract]",
"doi": "10.4153/CJM-1967-041-6",
"issn": "0008-414X",
"publisher": "Canadian Mathematical Society",
"publication": "Canadian Journal of Mathematics",
"publication_date": "1967-02",
"series_number": "1",
"volume": "19",
"issue": "1",
"pages": "488-498"
},
{
"id": "authors:nj55q-mkn11",
"collection": "authors",
"collection_id": "nj55q-mkn11",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-141551161",
"type": "article",
"title": "Archimedean quotient Riesz spaces",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "Moore Jr.",
"given_name": "L. C., Jr.",
"clpid": "Moore-L-C-Jr"
}
],
"abstract": "[no abstract]",
"doi": "10.1215/S0012-7094-67-03475-8",
"issn": "0012-7094",
"publisher": "Duke University Press",
"publication": "Duke Mathematical Journal",
"publication_date": "1967",
"series_number": "4",
"volume": "34",
"issue": "4",
"pages": "725-739"
},
{
"id": "authors:p7b7q-dwk51",
"collection": "authors",
"collection_id": "p7b7q-dwk51",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854579",
"type": "article",
"title": "Some examples of normed K\u00f6the spaces",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "Zaanen",
"given_name": "A. C.",
"clpid": "Zaanen-A-C"
}
],
"abstract": "Let X be a non-empty point set, and \u03bc a countably additive and non-negative measure in X. We assume that the Carath\u00e9odory extension procedure has already been applied to \u03bc, so that the \u03c3-field \u039b on which \u03bc is defined cannot be enlarged by another application of the Carath\u00e9odory procedure. Furthermore, it will be assumed that \u03bc is (totally) (\u03c3-finite, i.e., X is the union of a finite or countable number of sets of finite measure. Hence, the triple (X, \u039b, \u03bc) is a (totally) \u03c3-finite measure space in the usual terminology. The notation \u222b d \u03bc will denote integration (with respect to \u03bc) over the whole set X, and \u03c7 E = \u03c7 E (x) will stand for the characteristic function of the set E \u2282 X.",
"doi": "10.1007/bf01369107",
"issn": "0025-5831",
"publisher": "Springer Verlag",
"publication": "Mathematische Annalen",
"publication_date": "1966-10",
"series_number": "3",
"volume": "162",
"issue": "3",
"pages": "337-350"
},
{
"id": "authors:cvxy3-d6y83",
"collection": "authors",
"collection_id": "cvxy3-d6y83",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854393",
"type": "book_section",
"title": "Some Examples of Normed K\u00f6the Spaces",
"book_title": "Contributions to Functional Analysis",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "Zaanen",
"given_name": "A. C.",
"clpid": "Zaanen-A-C"
}
],
"abstract": "Let X be a non-empty point set, and \u03bc a countably additive and non-negative measure in X. We assume that the Carath\u00e9odory extension procedure has already been applied to \u03bc, so that the \u03c3-field \u039b on which \u03bc is defined cannot be enlarged by another application of the Carath\u00e9odory procedure. Furthermore, it will be assumed that \u03bc is (totally) (\u03c3-finite, i.e., X is the union of a finite or countable number of sets of finite measure. Hence, the triple (X, \u039b, \u03bc) is a (totally) \u03c3-finite measure space in the usual terminology. The notation \u222b d \u03bc will denote integration (with respect to \u03bc) over the whole set X, and \u03c7 E = \u03c7 E (x) will stand for the characteristic function of the set E \u2282 X.",
"doi": "10.1007/978-3-642-85997-7_22",
"isbn": "9783642859991",
"publisher": "Springer",
"place_of_publication": "Berlin",
"publication_date": "1966",
"pages": "337-350"
},
{
"id": "authors:a37xm-vdm93",
"collection": "authors",
"collection_id": "a37xm-vdm93",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-143713456",
"type": "article",
"title": "A remark on Sikorski's extension theorem for homomorphisms in the theory of Boolean algebras",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "[no abstract]",
"doi": "10.4064/fm-55-3-239-247",
"issn": "0016-2736",
"publisher": "Institute of Mathematics, Polish Academy of Sciences",
"publication": "Fundamenta Mathematicae",
"publication_date": "1964",
"volume": "55",
"pages": "239-247"
},
{
"id": "authors:5k5g6-hdy79",
"collection": "authors",
"collection_id": "5k5g6-hdy79",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-144613726",
"type": "article",
"title": "On Finitely Additive Measures in Boolean Algebras",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "[no abstract]",
"doi": "10.1515/crll.1964.213.165",
"issn": "1435-5345",
"publisher": "De Gruyter",
"publication": "Journal f\u00fcr die reine und angewandte Mathematik",
"publication_date": "1964",
"volume": "213",
"pages": "165-173"
},
{
"id": "authors:m9pqb-ztc94",
"collection": "authors",
"collection_id": "m9pqb-ztc94",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854668",
"type": "article",
"title": "Compactness of integral operators in Banach function spaces",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "Zaanen",
"given_name": "A. C.",
"clpid": "Zaanen-A-C"
}
],
"abstract": "[no abstract]",
"doi": "10.1007/bf01349240",
"issn": "0025-5831",
"publisher": "Springer Verlag",
"publication": "Mathematische Annalen",
"publication_date": "1963-04",
"series_number": "2",
"volume": "149",
"issue": "2",
"pages": "150-180"
},
{
"id": "authors:5cgv5-17k04",
"collection": "authors",
"collection_id": "5cgv5-17k04",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-145207989",
"type": "article",
"title": "A Property of the Fourier Coefficients of an Integrable Function",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "[no abstract]",
"doi": "10.2307/2312535",
"issn": "0002-9890",
"publisher": "Mathematical Association of America",
"publication": "American Mathematical Monthly",
"publication_date": "1962-02",
"series_number": "2",
"volume": "69",
"issue": "2",
"pages": "94-98"
},
{
"id": "authors:24mm1-zbt22",
"collection": "authors",
"collection_id": "24mm1-zbt22",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-145540187",
"type": "article",
"title": "Two applications of the method of construction by ultrapowers to anaylsis",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "[no abstract]",
"doi": "10.1090/S0002-9904-1962-10824-6",
"issn": "0273-0979",
"publisher": "American Mathematical Society",
"publication": "Bulletin of the American Mathematical Society",
"publication_date": "1962",
"series_number": "4",
"volume": "68",
"issue": "4",
"pages": "416-419"
},
{
"id": "authors:pv2ek-9jy26",
"collection": "authors",
"collection_id": "pv2ek-9jy26",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854757",
"type": "article",
"title": "Numerical methods and existence theorems for ordinary differential equations",
"author": [
{
"family_name": "Hull",
"given_name": "T. E.",
"clpid": "Hull-T-E"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "[no abstract]",
"doi": "10.1007/bf01386206",
"issn": "0029-599X",
"publisher": "Springer",
"publication": "Numerische Mathematik",
"publication_date": "1960-12",
"series_number": "1",
"volume": "2",
"issue": "1",
"pages": "30-41"
},
{
"id": "authors:mnkcz-9bc29",
"collection": "authors",
"collection_id": "mnkcz-9bc29",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181003-155854862",
"type": "article",
"title": "A remark on R. R. PHELPS' paper \"subreflexive normed linear spaces\"",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "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.",
"doi": "10.1007/bf01236931",
"issn": "0003-889X",
"publisher": "Springer",
"publication": "Archiv der Mathematik",
"publication_date": "1960-12",
"series_number": "1",
"volume": "11",
"issue": "1",
"pages": "192-193"
},
{
"id": "authors:w856h-y4v13",
"collection": "authors",
"collection_id": "w856h-y4v13",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-150532893",
"type": "article",
"title": "On the convergence of successive approximations in the theory of ordinary differential equations",
"author": [
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "[no abstract]",
"doi": "10.4153/CMB-1958-003-5",
"issn": "0008-4395",
"publisher": "CMS",
"publication": "Canadian Mathematical Bulletin",
"publication_date": "1958-01",
"series_number": "1",
"volume": "1",
"issue": "1",
"pages": "9-20"
},
{
"id": "authors:ebern-y4c21",
"collection": "authors",
"collection_id": "ebern-y4c21",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20181009-151135685",
"type": "article",
"title": "Reflexivity of the length function",
"author": [
{
"family_name": "Halperin",
"given_name": "Israel",
"clpid": "Halperin-I"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
}
],
"abstract": "[no abstract]",
"doi": "10.1090/S0002-9939-1957-0087903-X",
"issn": "0002-9939",
"publisher": "American Mathematical Society",
"publication": "Proceedings of the American Mathematical Society",
"publication_date": "1957",
"volume": "8",
"pages": "496-499"
}
]