Research On Isosceles Orthogonality And DSS Orthogonality

Generalized orthogonality is an important tool in the study of normed linear space geometry.The Isosceles orthogonality is one of the most basic generalized orthogonality,and the homogeneity plays a key role in characterizing the properties of its space.However,DSS orthogonality is another kind of generalized orthogonality different from Isosceles orthogonality,which has been given little attention.Based on the previous researches,we makes a further study of the properties of Isosceles orthogonality and DSS orthogonality.The main contents are as follows:In Chapter 2.we define a set HX? which related to Isosceles orthogonality,and respectively explored the relations among set HX? and set H'X,isometric reflection vector and L2-summable vector.In Chapter 3,we firstly proved that if ?>0,u?HX?,z?BX,z?Iu,then there exists a strictly decreasing and goes to zero sequence {?n}n=1+?(?)(0,1)which results u?I?nz,(?)n?N.Secondly,we studied the Birkhoff orthogonality and Isosceles orthogonality and their relations.According to the different ranges of ?,we studied the related properties of sets HX?.Finally,it is proved that if HX1? and the norm on X is almost transitive,then X is an inner product space.In chapter 4,we studied the geometric meaning and basic properties of DSS orthogonality,and prove that xl+y is equivalent to x?+y.The study of the relations between DSS orthogonality and Pythagorean orthogonality proved that there exists an inner product space.