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Constructive algebraic integration theory without choice”, in Mathematics, Algorithms and Proofs
 Dagstuhl Seminar Proceedings, 05021, Internationales Begegnungs und Forschungszentrum (IBFI), Schloss Dagstuhl
, 2005
"... Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpret ..."
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Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpreted in MartinL type theory or in a predicative topos in the sense of Moerdijk and Palmgren. We outline how to develop most of Bishop’s theorems on integration theory that do not mention points explicitly. Coquand’s constructive version of the Stone representation theorem is an important tool in this process. It is also used to give a new proof of Bishop’s spectral theorem.
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
PDF hosted at the Radboud Repository of the Radboud University
, 2002
"... Please be advised that this information was generated on 20160517 and may be subject to change. Locating the range of an operator with an adjoint ..."
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Please be advised that this information was generated on 20160517 and may be subject to change. Locating the range of an operator with an adjoint