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DOI: 10.4230/OASIcs.CCA.2009.2259

A Constructive Study of Landau's Summability Theorem

Authors: Josef Berger and Douglas Bridges

Published in: OASIcs, Volume 11, 6th International Conference on Computability and Complexity in Analysis (CCA'09) (2009)

Abstract

A summability theorem of Landau, which classically is a simple consequence of the uniform boundedness theorem, is examined constructively.

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Josef Berger and Douglas Bridges. A Constructive Study of Landau's Summability Theorem. In 6th International Conference on Computability and Complexity in Analysis (CCA'09). Open Access Series in Informatics (OASIcs), Volume 11, pp. 61-70, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{berger_et_al:OASIcs.CCA.2009.2259,
  author =	{Berger, Josef and Bridges, Douglas},
  title =	{{A Constructive Study of Landau's Summability Theorem}},
  booktitle =	{6th International Conference on Computability and Complexity in Analysis (CCA'09)},
  pages =	{61--70},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-12-5},
  ISSN =	{2190-6807},
  year =	{2009},
  volume =	{11},
  editor =	{Bauer, Andrej and Hertling, Peter and Ko, Ker-I},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.CCA.2009.2259},
  URN =		{urn:nbn:de:0030-drops-22595},
  doi =		{10.4230/OASIcs.CCA.2009.2259},
  annote =	{Keywords: Constructive analysis, Landau's theorem, uniform boundedness theorem Constructive analysis, Landau's theorem, uniform boundedness theorem}
}

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DOI: 10.4230/DagSemProc.04351.9

Compactness in apartness spaces?

Authors: Douglas Bridges, Hajime Ishihara, Peter Schuster, and Luminita S. Vita

Published in: Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

Abstract

A major problem in the constructive theory of apartness spaces is that of finding a good notion of compactness. Such a notion should (i) reduce to ``complete plus totally bounded'' for uniform spaces and (ii) classically be equivalent to the usual Heine-Borel-Lebesgue property for the apartness topology. The constructive counterpart of the smallest uniform structure compatible with a given apartness, while not constructively a uniform structure, offers a possible solution to the compactness-definition problem. That counterpart turns out to be interesting in its own right, and reveals some additional properties of an apartness that may have uses elsewhere in the theory.

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Douglas Bridges, Hajime Ishihara, Peter Schuster, and Luminita S. Vita. Compactness in apartness spaces?. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{bridges_et_al:DagSemProc.04351.9,
  author =	{Bridges, Douglas and Ishihara, Hajime and Schuster, Peter and Vita, Luminita S.},
  title =	{{Compactness in apartness spaces?}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--7},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4351},
  editor =	{Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.9},
  URN =		{urn:nbn:de:0030-drops-1175},
  doi =		{10.4230/DagSemProc.04351.9},
  annote =	{Keywords: Apartness , constructive , compact uniform space}
}

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Documents authored by Bridges, Douglas

A Constructive Study of Landau's Summability Theorem

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Compactness in apartness spaces?

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