Orthogonality in normed linear spaces: A classification of the different concepts and some open problems
Carlos Alberto Velasco Benítez · Revista Matemática Complutense · 1989
Orthogonality in normed linear spaces: a class¿fication of the dtfJ'erent concepis and sorne open problerns CARLOS BENíTEZABSTRAer.Orthogonality in inner product tpaces is a binary relation that can be expressed in many ways without explicit mention to the inner product of the space.Oreat pan of such definitions have alto tense in normed linear spaces.This simple observation it at the bate of ntany concepts of orthogonality in thete more general structuret.Various authors introduced such concepts ayer the last fifty years, although the origins of sorne of the mott interesting retults that can be obtained for these generalized concepts are, as usual in Mathematics, in previous or parallel works about convex sets, elliptis Gr elliptoids, duality, etc. (see, e.g., Gruber's paper [13]about the prior contributions of Caratheodory, Blaschke Gr Radon). DIFFERENT CONCEPTS OF ORTHOGONALITYAccording to its greater frecuency in tite literature this exposition is lirnited to tite case in wicit E is a real normed linear space.When tite norm of E is induced by an inner product, the ortitogonality of two points ir and y ob E is equivalent to each one of next (main or secondary) propositions.In tite more general context of normed linear spaces any one of such propositions is a definition of ot-thogonality between ir andy.