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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.
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...
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
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...