Study On Symmetry And Orthogonal Sets For Birkhoff Orthogonality

Studying the theory of generalized orthogonality in normed spaces is an invaluable way to explore and study the geometric properties of general normed spaces.Among many generalized orthogonalities,Birkhoff orthogonality plays a vital role in the geometric research of normed spaces.In recent years,the research on Birkhoff orthogonal symmetry has been the focus of many scholars,and some widely used research results have been obtained.At the same time,according to the generalized orthogonality of the normed space,the related theoretical knowledge of orthogonal sets in an inner product space can be extended to a general normed space.Furthermore,the related knowledge system of generalized orthogonal sets in a general normed space can be established.The primary research components of this paper are as follows:Firstly,we study the symmetry of Birkhoff orthogonality from a novel point of view,namely the mutual Birkhoff orthogonality in normed spaces.Prove the existence of mutually Birkhoff orthogonal elements of any non-zero element in a two-dimensional normed space.Give a sufficient condition for any two nonzero elements in a normed space that satisfy mutual Birkhoff orthogonality.We also prove that the mutual Birkhoff orthogonality and Roberts orthogonality are equivalent in l12 and l∞2.Secondly,inspired by the definition of orthogonal sets in inner product spaces,we put forward the concepts of Birkhoff orthogonal sets and mutually Birkhoff orthogonal sets.According to some examples,we show that Birkhoff orthogonal sets and mutually Birkhoff orthogonal sets are not necessarily linearly independent sets in general normed spaces.On this basis,we further prove that Birkhoff orthogonal sets are linearly independent sets in two-dimensional smooth or strictly convex real normed spaces;mutually Birkhoff orthogonal sets are linearly independent sets in finite-dimensional smooth real normed spaces;Birkhoff orthogonal sets containing left(right)symmetric points are linearly independent sets in three-dimensional real normed spaces that are both smooth and strictly convex.Finally,we extend the study of Birkhoff orthogonal sets to normed spaces with higher dimensions.Prove the existence of Birkhoff orthonormal basis with special properties in n-dimensional real normed spaces.According to the existing results,it is shown that the linear independence of Birkhoff orthogonal sets is connected with the smoothness of the space in which they are located.The properties of Birkhoff orthogonal sets are studied in n-dimensional smooth real normed spaces.