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Brouwer's fan theorem and unique existence in constructive analysis

✍ Scribed by Josef Berger; Hajime Ishihara

Publisher: John Wiley and Sons
Year: 2005
Tongue: English
Weight: 92 KB
Volume: 51
Category: Article
ISSN: 0044-3050
DOI: 10.1002/malq.200410038

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✦ Synopsis

Many existence propositions in constructive analysis are implied by the lesser limited principle of omniscience LLPO; sometimes one can even show equivalence. It was discovered recently that some existence propositions are equivalent to Bouwer's fan theorem FAN if one additionally assumes that there exists at most one object with the desired property. We are providing a list of conditions being equivalent to FAN, such as a unique version of weak König's lemma. This illuminates the relation between FAN and LLPO. Furthermore, we give a short and elementary proof of the fact that FAN is equivalent to each positive valued function with compact domain having positive infimum.

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## Abstract We prove the Banach‐Steinhaus theorem for distributions on the space 𝒟(ℝ) within Bishop's constructive mathematics. To this end, we investigate the constructive sequential completion $ \tilde {\cal D} $(ℝ) of 𝒟(ℝ).