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Sequential continuity of linear mappings in constructive mathematics
 J. Universal Computer Science
, 1997
"... Abstract: This paper deals, constructively, with two theorems on the sequential continuity of linear mappings. The classical proofs of these theorems use the boundedness of the linear mappings, which is a constructively stronger property than sequential continuity; and constructively inadmissable ve ..."
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Abstract: This paper deals, constructively, with two theorems on the sequential continuity of linear mappings. The classical proofs of these theorems use the boundedness of the linear mappings, which is a constructively stronger property than sequential continuity; and constructively inadmissable versions of the BanachSteinhaus theorem.
Sequentially Continuous Linear Mappings in Constructive Analysis
, 1996
"... this paper we derive some results about sequentially continuous linear mappings within BISH. These results tend to reinforce our hope that such mappings may turn out to be bounded (continuous) after all. For background material on BISH, see [1], and for information about the relation between BISH, I ..."
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this paper we derive some results about sequentially continuous linear mappings within BISH. These results tend to reinforce our hope that such mappings may turn out to be bounded (continuous) after all. For background material on BISH, see [1], and for information about the relation between BISH, INT, and RUSS, see [2]. 2 Sequential continuity preserves Cauchyness
MEANING IN CLASSICAL MATHEMATICS: IS IT AT ODDS WITH INTUITIONISM?
, 2011
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On the foundations of constructive mathematics – especially in relation to the theory of continuous functions
 Foundations of Science
"... We discuss the foundations of constructive mathematics, including recursive mathematics and intuitionism, in relation to classical mathematics. There are connections with the foundations of physics, due to the way in which the different branches of mathematics reflect reality. Many different axioms ..."
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We discuss the foundations of constructive mathematics, including recursive mathematics and intuitionism, in relation to classical mathematics. There are connections with the foundations of physics, due to the way in which the different branches of mathematics reflect reality. Many different axioms and their interrelationship are discussed. We show that there is a fundamental problem in bish (Bishop’s school of constructive mathematics) with regard to its current definition of ‘continuous function’. This problem is closely related to the definition in bish of ‘locally compact’. Possible approaches to this problem are discussed. Topology seems to be a key to understanding many issues. We offer several new simplifying axioms, which can form bridges between the various branches of constructive mathematics and classical mathematics (‘reuniting the antipodes’). We give a simplification of basic intuitionistic theory, especially with regard to socalled ‘bar induction’. We then plead for a limited number of axiomatic systems, which differentiate between the various branches of mathematics. Finally, in the appendix we offer bish an elegant topological definition of ‘locally compact’, which unlike the current definition is equivalent to the usual classical and/or intuitionistic definition in classical and intuitionistic mathematics respectively.
Constructive Closed Range and Open Mapping Theorems
 Indag. Math. N.S
, 1998
"... We prove a version of the closed range theorem within Bishop's constructive mathematics. This is applied to show that if an operator T on a Hilbert space has an adjoint and a complete range, then both T and T are sequentially open. 1 Introduction In this paper we continue our constructiv ..."
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We prove a version of the closed range theorem within Bishop's constructive mathematics. This is applied to show that if an operator T on a Hilbert space has an adjoint and a complete range, then both T and T are sequentially open. 1 Introduction In this paper we continue our constructive exploration of the theory of operators, in particular operators on a Hilbert space ([4], [5], [6]). We work entirely within Bishop's constructive mathematics, which we regard as mathematics with intuitionistic logic. For discussions of the merits of this approach to mathematics in particular, the multiplicity of its modelssee [3] and [10]. The technical background needed in our paper is found in [1] and [9]. Our main aim is to prove the following result, the constructive Closed Range Theorem for operators on a Hilbert space (cf. [16], pages 99103): Theorem 1 Let H be a Hilbert space, and T a linear operator on H such that T exists and ran(T ) is closed. Then ran(T ) and ker(T ) are bo...
Bounded variation implies regulated: A constructive proof. The Journal of symbolic logic
"... Abstract. It is shown constructively that a strongly extensional function of bounded variation on an interval is regulated, in a sequential sense that is classically equivalent to the usual one. This paper continues the constructive study of monotone functions and functions of bounded variation, beg ..."
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Abstract. It is shown constructively that a strongly extensional function of bounded variation on an interval is regulated, in a sequential sense that is classically equivalent to the usual one. This paper continues the constructive study of monotone functions and functions of bounded variation, begun in [3] and [4] (see also [5]). It can be read by anyone who appreciates the distinction between classical and intuitionistic logic, and does not require a detailed knowledge of the constructive theory of R, let alone any abstract constructive analysis. However, the reader will find it helpful to have at hand a copy of [1], [2], [6], or [10]. Throughout the paper, I will be a proper interval in R, Y a metric space,1 and f: I → Y a mapping that is strongly extensional in the sense that ∀x∀y (f(x) 6 = f(y) ⇒ x 6 = y), where, for two elements x, y of a metric space, x 6 = y means ρ(x, y)> 0. It is shown in [4] that an increasing function f: I → R is strongly extensional, and that for all applicable x ∈ I the real numbers f(x−) = limt→x − f(x) and f(x+) = limt→x+ f(t)
The inversion problem for computable linear operators
 OF LECT. NOT. COMP. SCI
, 2003
"... Given a program of a linear bounded and bijective operator T, does there exist a program for the inverse operator T −1? And if this is the case, does there exist a general algorithm to transfer a program of T into a program of T −1? This is the inversion problem for computable linear operators on ..."
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Given a program of a linear bounded and bijective operator T, does there exist a program for the inverse operator T −1? And if this is the case, does there exist a general algorithm to transfer a program of T into a program of T −1? This is the inversion problem for computable linear operators on Banach spaces in its nonuniform and uniform formulation, respectively. We study this problem from the point of view of computable analysis which is the Turing machine based theory of computability on Euclidean space and other topological spaces. Using a computable version of Banach’s Inverse Mapping Theorem we can answer the first question positively. Hence, the nonuniform version of the inversion problem is solvable, while a topological argument shows that the uniform version is not. Thus, we are in the striking situation that any computable linear operator has a computable inverse while there exists no general algorithmic procedure to transfer a program of the operator into a program of its inverse. As a consequence, the computable version of Banach’s Inverse Mapping Theorem is a powerful tool which can be used to produce highly nonconstructive existence proofs of algorithms. We apply this method to prove that a certain initial value problem admits a computable solution.
Various Continuity Properties in Constructive Analysis
"... . This paper deals with continuity properties of functions f : X ! Y between metric spaces X and Y within the framework of Bishop's constructive mathematics. We concentrate on the situation when X is not complete. We investigate the relations between the properties of nondiscontinuity, seque ..."
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. This paper deals with continuity properties of functions f : X ! Y between metric spaces X and Y within the framework of Bishop's constructive mathematics. We concentrate on the situation when X is not complete. We investigate the relations between the properties of nondiscontinuity, sequential continuity, mapping Cauchy sequences to totally bounded sequences, and a certain boundedness condition. 1. Introduction In constructive mathematics, investigations into conditions that ensure the continuity of a function from one metric space to another go back at least to Brouwer, who proved that every function from the real numbers to a metric space must be continuous [3, 4]. In recursive constructive mathematics, Markov showed, in 1954, that every function f : R ! R is nondiscontinuous 1 . Tsejtin [20], and Kreisel, Lacombe and Shoeneld [14] extended this to show that every function of a complete separable metric space into a separable metric space is continuous. Orevkov [16] pro...
Constructive aspects of Riemanns permutation theorem for series
, 2012
"... The notions of permutable and weakpermutable convergence of a seriesP1 n=1 an of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemanns two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely ..."
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The notions of permutable and weakpermutable convergence of a seriesP1 n=1 an of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemanns two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishopstyle constructive mathematics, we prove that Ishiharas principle BDN implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation theorem for series holds but BDN does not, the best we can hope for as a partial converse to our
rst theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies BDN. We show that this is the case when the property is weakpermutable convergence. 1