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 Banach-Steinhaus theorem.

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 models|see [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 99-103): 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...

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

Abstract. Given a program of a linear bounded and bijective operator T, does there exist a program for the inverse operator T −1?Andif 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 non-uniform 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 non-uniform 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 non-constructive existence proofs of algorithms. We apply this method to prove that a certain initial value problem admits a computable solution.