Othogonality In Banach Space

Abstract. The theory of orthogonality in Hilbert space is one of the most important contents of functional analysis. The orthogonal projections, orthogonal bases, orthogonal complements and so on, induced by orthogonality, not only rich Hilbert space theory, but also characterize geometrical characters of Hilbert space. Orthogonality concept in Banach space is a natural generalization of orthogonality in Hilbert space and an important development of Banach space theory. In this thesis, we discuss some important concepts of orthogonalities in Banach space and give some characterizations of them. We also establish some relationships between them. This paper is divided into three chapters and the main results are given as follows.In Chapter 1, we study Birkhoff orthogonality which is the most widely used one. First, we give the relation of Birkhoff orthogonality and functionals in Banach space, then use functionals as a tool to investigate the relations of Birkhoff orthogonality and underling Banach space. In Banach spaces l_S~P and C_p, we give some properties Birkhoff orthogonality. In light of orthogonality, we define and discuss orthogonal complement of a set in a Banach space. Lastly, we investigate some properties of the Birkhoff orthogonal complements.In Chapter 2, we study Isosceles orthogonality and Pythagorean orthogonality, which are natural extensions of orthogonality in Hilbert space. We first introduce Isosceles orthogonality. By its definition, we know it has consanguineous relation with isometry, thus we can study isometry to investigate it. The emphase of the chapter is to study Isosceles orthogonal complementary, and use it to study geometrical properties of Banach space. Last, we study the properties of Pythagorean orthogonality.In Chapter 3, we discuss the other orthogonalities in in Banach space, and give some properties of them. The emphase of the chapter is to study the relations of Birkhoff orthogonality, Isosceles orthogonality and Pythagorean orthogonality, and it is proved that if one of these induces another, then the space is an inner product space.