Abstract
In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations.
Highlights
During the last years many papers concerning various aspects of orthogonalities in the field of functional equations and inequalities have appeared
As long as we are working in inner product spaces, usually there is no doubt what kind of orthogonality relation we have in mind
We start the survey with listing some orthogonality relations described in normed spaces
Summary
During the last years many papers concerning various aspects of orthogonalities in the field of functional equations and inequalities have appeared. In this paper we want to give some overview on these results as well as to collect a number of items from the literature dealing with the subject. It is worth mentioning papers by Paganoni and Ratz [139] from 1995, Ratz [156] from 2001 and Chmielinski [42,44] from 2006, 2012, respectively, where the reader can find some partial collections of the results in this domain
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