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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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Cited by 4 (3 self)
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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...
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 (e ..."
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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...
Locating the range of an operator on a Hilbert space
 Bull. London Math. Soc
, 1992
"... In classical operator theory it is taken for granted that we can project onto the closure of the range of an operator T on a Hilbert space H. In a constructive development of operator theory, to which this note is a contribution, this projection exists if and only if ran(r), the range of T, is locat ..."
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In classical operator theory it is taken for granted that we can project onto the closure of the range of an operator T on a Hilbert space H. In a constructive development of operator theory, to which this note is a contribution, this projection exists if and only if ran(r), the range of T, is located, in the sense that the distance
LOCATING SUBSETS OF A HILBERT SPACE HAJIME ISHIHARA
"... Abstract. 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. ..."
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Abstract. 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.