Orthogonality in Normed Linear Spaces

Abstract

This thesis deals with the orthogonality in normed linear spaces. The goal is to investigate and study different notions of orthogonality in normed spaces. By utilizing the 2-HH norm and bounded linear operators, some notions of orthogonality are introduced and then, different properties of orthogonality in relation to these orthogonalities are studied. We generalize the Robert, Birkhoff-James, and a new orthogonality in terms of the 2HH norm, and study the main properties of orthogonality. We prove that the Birkoff and Robert orthogonality in terms of the 2-HH norm are equivalent if the underlying space is real inner product space. Further, we prove that the isosceles orthogonality is homogeneous if and only if it is additive. Additionally, we prove that the orthogonality relation of type (I) in terms of 2-HH norm satisfies non-degeneracy, simplification, continuity, and uniqueness properties. Moreover, we prove that the Carlsson orthogonality in terms of bounded linear operators also satisfies non-degeneracy, simplification, and continuity properties. In the case of norm attaining bounded linear operator with disjoint support in a Hilbert space H, we prove that two operators are orthogonal in the sense of Pythagoras if and only if they are orthogonal in the sense of isosceles. In terms of buonded linear operators, we prove that the Pythagorean orthogonality and orthogonality relation of type (I), imply the Birkhoff-James orthogonality, but the converse may not be true. Under the restriction of an element belonging to the norm attainment set, we prove that the orthogonality of images also implies the orthogonality of operators in the Carlsson as well as Robert’s sense. Finally, as applications, we prove that the Pythagorean orthogonality implies the best approximation, and the best approximation (resp. best approximation) and Birkhoff orthogonality ( resp. Birkhoff orthogonality) are equivalents. Keywords: Normed linear spaces, Inner-product space, Birkhoff-James orthogonality, Pythagorean orthogonality, p-HH norm, Best approximation