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The constructive completion of the space (ℝ)

✍ Scribed by Satoru Yoshida

Publisher
John Wiley and Sons
Year
2005
Tongue
English
Weight
115 KB
Volume
51
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

We prove in the framework of Bishop's constructive mathematics that the sequential completion \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \tilde {\cal D} $\end{document}(ℝ) of the space 𝒟(ℝ) is filter‐complete. Then it follows as a corollary that the filter‐completeness of 𝒟(ℝ) is equivalent to the principle BD‐ℕ, which can be proved in classical mathematics, Brouwer's intuitionistic mathematics and constructive recursive mathematics of Markov's school, but does not in Bishop's constructive mathematics. We also show that \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \tilde {\cal D} $\end{document}(ℝ) is identical with the filter‐completion which was provided by Bishop. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

📜 SIMILAR VOLUMES

The Banach-Steinhaus theorem for the spa
The Banach-Steinhaus theorem for the space (ℝ) in constructive analysis
✍ Satoru Yoshida 📂 Article 📅 2003 🏛 John Wiley and Sons 🌐 English ⚖ 197 KB

## 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).