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Locating Subsets Of A Hilbert Space
 Proceedings of the American Mathematical Society, 129(5):1385–1390, 2001. B.: Constructive Results on Operator Algebras
, 1998
"... . This paper deals with locatedness of convex subsets in inner product and Hilbert spaces which plays a crucial role in the constructive validity of many important theorems of analysis. 1. Introduction In Bishop's constructive mathematics, the framework of this paper, locatedness of subsets (especi ..."
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. This paper deals with locatedness of convex subsets in inner product and Hilbert spaces which plays a crucial role in the constructive validity of many important theorems of analysis. 1. Introduction In Bishop's constructive mathematics, the framework of this paper, locatedness of subsets (especially convex subsets) of normed spaces plays a crucial role in the validity of many important theorems of analysis such as the HahnBanach and separation theorems [1, Chapter 7.4], [7], the open and unopen mapping theorems [5], and the existence theorems of Minkowski functionals [9]. (Recall that subset C of a normed space X is located if (x; C) := inffkx yk : y 2 Cg exists for each x 2 X.) Richman [10] extended the denition of weakly totally boundedness, which was rst dened in [8] for separable Hilbert spaces, to inner product spaces which are not necessarily separable as follows: a subset C of an inner product space X is weakly totally bounded if for each x 2 X , fhx; yi : y 2 Cg i...
Cauchy nets in the constructive theory of apartness spaces
 Scientiae Mathematicae Japonicae
, 2002
"... Abstract. A notion of Cauchy net is introduced into the constructive theory of apartness spaces. It is shown that for a sequence in a metric space this notion is equivalent to the standard metric notion of Cauchy sequence. Applications of this notion are then given, culminating in a generalisation o ..."
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Abstract. A notion of Cauchy net is introduced into the constructive theory of apartness spaces. It is shown that for a sequence in a metric space this notion is equivalent to the standard metric notion of Cauchy sequence. Applications of this notion are then given, culminating in a generalisation of Bishop’s Lemma on locatedness. 1 Introduction Axioms for a constructive theory of apartness between sets were introduced in [12], where the particular example of a uniform space was discussed in detail. In the present paper we discuss Cauchy and convergent sequences in the framework of that theory. By constructive mathematics we mean mathematics developed with intuitionistic logic
Constructive Order Completeness
, 2004
"... Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics. Unlike the classical case, one cannot prove constructively that every nonempty bounded above set of real numbers has a supremum. However, the order completeness of R is expressed constructively by an ..."
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Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics. Unlike the classical case, one cannot prove constructively that every nonempty bounded above set of real numbers has a supremum. However, the order completeness of R is expressed constructively by an equivalent condition for the existence of the supremum, a condition of (upper) order locatedness which is vacuously true in the classical case. A generalization of this condition will provide a definition of upper locatedness for a partially ordered set. It turns out that the supremum of a set S exists if and only if S is upper located and has a weak supremum—that is, the classical least upper bound. A partially ordered set will be called order complete if each nonempty subset that is bounded above and upper located has a supremum. It can be proved that, as in the classical mathematics, R n is order complete. 1
A Constructive Theory of PointSet Nearness
 in Proceedings of Topology in Computer Science: Constructivity; Asymmetry and Partiality; Digitization, Seminar in Dagstuhl, Germany, 4–9 June 2000; Springer Lecture Notes in Computer Science
, 2001
"... An axiomatic constructive development of the theory of nearness and apartness of a point and a set is introduced as a setting for topology. ..."
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An axiomatic constructive development of the theory of nearness and apartness of a point and a set is introduced as a setting for topology.
Deciding LinearTranscendental Problems
, 2000
"... We present a decision procedure for lineartranscendental problems formalized in a suitable firstorder language. The problems are formalized by formulas with arbitrary quantified linear variables and a block of quantifiers with respect to mixed lineartranscendental variables. Variables may range b ..."
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We present a decision procedure for lineartranscendental problems formalized in a suitable firstorder language. The problems are formalized by formulas with arbitrary quantified linear variables and a block of quantifiers with respect to mixed lineartranscendental variables. Variables may range both over the reals and over the integers. The transcendental functions admitted are characterized axiomatically; they include the exponential function applied to a polynomial, hyperbolic functions and their inverses, and the arcustangent. The decision procedure is explicit and implementable; it is based on mixed realinteger linear elimination, the symbolic test point method, elementary analysis, and Lindemann's theorem. As a byproduct we obtain sample solutions for existential formulas and a qualitative description of the connected components of the satisfaction set wrt. a mixed lineartranscendental variable.
PLURALISM IN MATHEMATICS
, 2004
"... We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic. ..."
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We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic.
Vitali’s theorem and WWKL
 Archive for Mathematical Logic
"... Abstract. Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA0. We show that several wellknown measuretheoretic propositions ..."
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Abstract. Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA0. We show that several wellknown measuretheoretic propositions including the Vitali Covering Theorem are equivalent to WWKL over RCA0. 1.
Adjoints, absolute values and polar decompositions
 Journal of Operator Theory
"... Abstract. Various questions about adjoints, absolute values and polar decompositions of operators are addressed from a constructive point of view. The focus is on bilinear forms. Conditions are given for the existence of an adjoint, and a general notion of a polar decomposition is developed. The Rie ..."
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Abstract. Various questions about adjoints, absolute values and polar decompositions of operators are addressed from a constructive point of view. The focus is on bilinear forms. Conditions are given for the existence of an adjoint, and a general notion of a polar decomposition is developed. The Riesz representation theorem is proved without countable choice.
Sets, Complements and Boundaries
"... The relations among a set, its complement, and its boundary are examined constructively. A crucial tool is a theorem that allows the construction of a point where a segment comes close to the boundary of a set in a Banach space. Brouwerian examples show that many of the results are the best possible ..."
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The relations among a set, its complement, and its boundary are examined constructively. A crucial tool is a theorem that allows the construction of a point where a segment comes close to the boundary of a set in a Banach space. Brouwerian examples show that many of the results are the best possible.
Constructive Aspects of the Dirichlet Problem 1
"... Abstract: We examine, within the framework of Bishop's constructive mathematics, various classical methods for proving the existence of weak solutions of the Dirichlet Problem, with a view to showing why those methods do not immediately translate into viable constructive ones. In particular, we disc ..."
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Abstract: We examine, within the framework of Bishop's constructive mathematics, various classical methods for proving the existence of weak solutions of the Dirichlet Problem, with a view to showing why those methods do not immediately translate into viable constructive ones. In particular, we discuss the equivalence of the existence of weak solutions of the Dirichlet Problem and the existence of minimizers for certain associated integral functionals. Our analysis pinpoints exactly what is needed to nd weak solutions of the Dirichlet Problem: namely, the computation of either the norm of a linear functional on a certain Hilbert space or, equivalently, the in mum of an associated integral functional.