Generalized Orthogonal Groups In Normed Linear Spaces And Related Properties

The problem of orthogonality is actually an important research content of inner product space.For a long time,as scholars have deepened their understanding of functional analysis,especially Banach geometric theory.Generalized orthogonal theory in the normed linear space was established and some research were carried out,and corresponding results were obtained.Various essential characteristics of orthogonality were used to extend the orthogonality from the inner product space to the general normed linear space.On the other hand,orthogonal groups,orthogonal bases,orthogonalization of vector groups,etc.have been extensively studied,forming a complete content system to a certain extent,and the corresponding results have played an important role in the improvement of the inner product space theory.Naturally,it was a high time to extend related theories to general normed linear spaces.In addition,in the specific normed linear space,isometric reflection was used to describe some general orthogonal relations,so some specific properties of orthogonal in the specific normed linear space were obtained by introducing the isometric reflection vector.This paper first introduces the corresponding generalized isosceles orthogonal group,generalized Birkhoff orthogonal group,generalized Roberts orthogonal group,strong Birkhoff orthogonal group and the corresponding generalized orthogonal basis in the general normed linear space.It is proved that any two non-zero generalized orthogonal groups must be linearly independent groups.An example shows that there are four non-zero Birkhoff orthogonal groups in a two-dimensional space.This example allows us to pay attention to the research on the related properties of generalized orthogonal groups.Secondly,the relativity of generalized orthogonal groups and the Schmidt orthogonalization method of linear independent groups in specific space were discussed.Next,according to the corresponding properties of Birkhoff's orthogonality in two-dimensional normalized linear space and the research results of isometric reflection in specific normalized linear space,the specific application of Birkhoff orthogonality inl_p space is studied,and finally the isometric reflection is extended to other specific research in Orlicz sequence space.