caa a22 4500 388039329 CHVBK 20180307125016.0 cr unu---uuuuu 161130e199806 xx s 000 0 eng 10.2307/2586851 doi S0022481200015097 pii (NATIONALLICENCE)cambridge-10.2307/2586851 Sequentially continuous linear mappings in constructive analysis [Elektronische Daten] A mapping u: X → Y between metric spaces is sequentially continuous if for each sequence (xn ) converging to x ∈ X, (u(xn )) converges to u(x). It is well known in classical mathematics that a sequentially continuous mapping between metric spaces is continuous; but, as all proofs of this result involve the law of excluded middle, there appears to be a constructive distinction between sequential continuity and continuity. Although this distinction is worth exploring in its own right, there is another reason why sequential continuity is interesting to the constructive mathematician: Ishihara [8] has a version of Banach's inverse mapping theorem in functional analysis that involves the sequential continuity, rather than continuity, of the linear mappings; if this result could be upgraded by deleting the word "sequential”, then we could prove constructively the standard versions of the inverse mapping theorem and the closed graph theorem. Troelstra [9] showed that in Brouwer's intuitionistic mathematics (INT) a sequentially continuous mapping on a separable metric space is continuous. On the other hand, Ishihara [6, 7] proved constructively that the continuity of sequentially continuous mappings on a separable metric space is equivalent to a certain boundedness principle for subsets of ℕ; in the same paper, he showed that the latter principle holds within the recursive constructive mathematics (RUSS) of the Markov School. Since it is not known whether that principle holds within Bishop's constructive mathematics (BISH), of which INT and RUSS are models and which can be regarded as the constructive core of mathematics, the exploration of sequential continuity within BISH holds some interest. Copyright © Association for Symbolic Logic 1998 Bridges Douglas Department of Mathematics, University of Waikato, Hamilton, New Zealand, E-mail: douglas@waikato.ac.nz Mines Ray Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA, E-mail: ray@nmsu.edu The Journal of Symbolic Logic Cambridge University Press 63/2(1998-06), 579-583 0022-4812 63:2<579 1998 63 JSL https://doi.org/10.2307/2586851 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.2307/2586851 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bridges Douglas Department of Mathematics, University of Waikato, Hamilton, New Zealand, E-mail: douglas@waikato.ac.nz NATIONALLICENCE 700 1- Mines Ray Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA, E-mail: ray@nmsu.edu NATIONALLICENCE 773 0- The Journal of Symbolic Logic Cambridge University Press 63/2(1998-06), 579-583 0022-4812 63:2<579 1998 63 JSL CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge