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Journal of Mathematics
Volume 2017 (2017), Article ID 1073589, 9 pages
https://doi.org/10.1155/2017/1073589
Research Article

Orthogonal Symmetries and Reflections in Banach Spaces

University of Sharjah, Sharjah, UAE

Correspondence should be addressed to Ali Jaballah; ea.ca.hajrahs@hallabaja

Received 5 March 2017; Accepted 12 April 2017; Published 14 June 2017

Academic Editor: Ji Gao

Copyright © 2017 Ali Jaballah and Fathi B. Saidi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a Banach space. We introduce a concept of orthogonal symmetry and reflection in . We then establish its relation with the concept of best approximation and investigate its implication on the shape of the unit ball of the Banach space by considering sections over subspaces. The results are then applied to the space of continuous functions on a compact set . We obtain some nontrivial symmetries of the unit ball of . We also show that, under natural symmetry conditions, every odd function is orthogonal to every even function in . We conclude with some suggestions for further investigations.