Some Research On Orthogonalities In Normed Linear Spaces

Functional analysis is the important research fields in basic mathematics.The orthogonality is one of the directions of operator theory in functional analysis.Based on the existing orthogonality in normed linear space,the relationship between them is studied,and then several new orthogonalities in normed linear space are proposed.Based on the concept of orthogonal preserving mapping,several new orthogonal preserving mappings are introduced,their properties and their relations with several isometric mappings are studied.The main research results are shown as follows:1.The properties and relations of several orthogonalities in normed linear space and the properties in specific function space are studied.2.The concepts of（ε₁,ε₂）-S orthogonality and approximatee-I orthogonality are proposed.The relationship between（ε₁,ε₂）-S orthogonality preserving mapping and approximate（δ₁,δ₂）-isometric mapping and the relationship between approximatee-I orthogonal preserving mapping and generalizede-isometric preserving mapping are studied.A characterization of approximatee-I orthogonal preserving mapping is given.3.The concept of（ε₁,ε₂）-R orthogonal set in real normed linear space is proposed,and a characterization of the linearity of（ε₁,ε₂）-R orthogonal set is given.Then the concepts of（ε₁,ε₂）-R orthogonal preserving mapping and（ε₁,ε₂）-W orthogonal preserving mapping are proposed.The properties of these two orthogonal preserving mappings and the relations of approximate（δ₁,δ₂）-isometric mapping and generalizeδ（δ₁,δ₂）-isometric mapping with them are given.