Riesz's lemma and orthogonality in normed spaces (Q1084620)

A geometric interpretation of Riesz's lemma is given in terms of duality theory of normed spaces and two types of generalizations of Birkhoff orthogonality are presented. The first notion of orthogonality is formulated via the Bishop-Phelps theorem and a characterization of non- reflexivity of Banach spaces is obtained in connection with the James theorem. The second notion is based on the geometric meaning of Riesz's lemma and is adequate for treating orthogonality problems in incomplete normed spaces. An orthogonality theorem for proper closed subspaces of general normed spaces is established and nonreflexivity of the completions of incomplete normed spaces is discussed in some detail. Moreover, given a nonreflexive Banach space X, problems on the existence and properties of elements of \(X^{**}\) orthogonal to X in the sense of Birkhoff are studied from the point of view of the geometry of second dual Banach spaces: The set of left-orthogonal elements to X and the set of right-orthogonal elements to X are introduced and orthogonal decomposition in a generalized sense of \(X^{**}\) is considered by means of those sets. Finally, the structure of abstract L and M spaces is investigated through the results mentioned above.