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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...
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...
Locating the range of an operator with an adjoint
, 2002
"... In this paper we consider the following question: given a linear operator 1 on a Hilbert space, can we compute the projection on the closure of its range? Instead of making the notion of computation precise, we use Bishop's informal approach [1], in which `there exists ' is interpreted st ..."
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In this paper we consider the following question: given a linear operator 1 on a Hilbert space, can we compute the projection on the closure of its range? Instead of making the notion of computation precise, we use Bishop's informal approach [1], in which `there exists ' is interpreted strictly as `we can compute'. It turns out that the reasoning we use to capture this interpretation can be described by intuitionistic logic. This logic diers from classical logic by not recognising certain principles, such as the scheme `P or not P ', as generally valid. Since we do not adopt axioms that are classically false, all our theorems are acceptable in classical mathematics. To answer our initial question armatively, it is enough to show that the range ran (T) of the operator T on the Hilbert space H is locatedthat is, the distance (x; ran (T)) = inf fkx Tyk: y 2 Hg exists (is computable) for each x 2 H ([2], pages 366 and 371). The locatedness of the kernel ker (T ) of the adjoint T of T is easily seen to be a necessarybut according to Example 1 of [6], not sucientcondition for ran (T) to be located. Theorem gives necessary and sucient conditions under which the locatedness of ker (T ) ensures that