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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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Cited by 8 (6 self)
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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
Operator ranges, integrable sets, and the functional calculus
 Houston J. Math
, 1985
"... In this paper we consider some questions about the range of an operator on a Hilbert space, and the measure of the intersection of a decreasing sequence of integrable compact sets in a locally compact space. Classically, each of these questions has a trivial answer. However, within the framework of ..."
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Cited by 5 (3 self)
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In this paper we consider some questions about the range of an operator on a Hilbert space, and the measure of the intersection of a decreasing sequence of integrable compact sets in a locally compact space. Classically, each of these questions has a trivial answer. However, within the framework of Bishop's constructive mathematics, the attempt to answer them leads to some interesting results in operator theory and measure theory. These results are linked together by means of the functional calculus, to give a constructive criterion for the existence of the projection on the closure of the range of a Hermitian operator. Since there is a sharp contrast between the classical and constructive treatments of the questions we consider, we shall present our results and examples in some detail, in the hope that a casual reader unfamiliar with the constructive approach may gain some insight into recent developments in constructive mathematics and its methodology. Thus the expert in constructive mathematics may find more detail than he needs for an understanding of our results: for him, the main point of the paper will be Theorem (4.6), although some of the earlier mâ€¢terial, such as Proposition (3.2) and certain examples, will also be of interest. The reader unfamiliar with the foundations of constructive mathematics should consult [2, 3, 4]. The interested reader should also examine [5], which discusses the existence of the projection on the kernel of a bounded linear map into a finitedimensional space. NOTATION. For any mapping f, dmn f and ran f are respectively the domain and the range of f. If ran f is a vector space, then kerf is the kernel of f. 1. When does the pro/ection of a Hilbert space H on a closed littear subset exist? From the constructive viewpoint of this paper, this question is not as ridiculous
CONSTRUCTIVE COMPACT LINEAR MAPPINGS
"... In this paper, we deal with compact linear mappings of a normed linear space, within the framework of Bishop's constructive mathematics. We prove the constructive substitutes for the classically wellknown theorems on compact linear mappings: T is compact if and only if T * is compact; if S is ..."
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In this paper, we deal with compact linear mappings of a normed linear space, within the framework of Bishop's constructive mathematics. We prove the constructive substitutes for the classically wellknown theorems on compact linear mappings: T is compact if and only if T * is compact; if S is bounded and if T is compact, then TS is compact; if S and T is compact, then S+ T is compact. 1.