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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Computing infima on convex sets, with applications in Hilbert spaces


Authors: Douglas Bridges, Hajime Ishihara and Luminita Vîta
Journal: Proc. Amer. Math. Soc. 132 (2004), 2723-2732
MSC (2000): Primary 03F60, 46S30
Published electronically: April 8, 2004
MathSciNet review: 2054799
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Abstract: Using intuitionistic logic, we prove that under certain reasonable conditions, the infimum of a real-valued convex function on a convex set exists. This result is then applied to problems of simultaneous approximation in Hilbert space $H$ and the corresponding operator space $\mathcal{B}\left(H\right).$ This enables us to establish that a bounded, weak-operator totally bounded, convex subset of $\mathcal{B}\left( H\right)$ is strong-operator located.


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Additional Information

Douglas Bridges
Affiliation: Department of Mathematics and Statistics, Private Bag 4800, University of Canterbury, Christchurch, New Zealand
Email: d.bridges@math.canterbury.ac.nz

Hajime Ishihara
Affiliation: School of Information Science, Japan Advanced Institute of Science and Technology, Tatsunokuchi, Ishikawa 923-1292, Japan
Email: ishihara@jaist.ac.jp

Luminita Vîta
Affiliation: Department of Mathematics and Statistics, Private Bag 4800, University of Canterbury, Christchurch, New Zealand
Email: Luminita@math.net

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07496-9
PII: S 0002-9939(04)07496-9
Keywords: Functional analysis, constructive mathematics.
Received by editor(s): January 6, 2003
Published electronically: April 8, 2004
Additional Notes: Bridges and Vîţă gratefully acknowledge the support of the Marsden Fund and FoRST New Zealand. All three authors thank the Japan Advanced Institute of Science & Technology for supporting the visit by the first two during which much of this work was carried out.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2004 American Mathematical Society