Research On Problems Related To The Existence Of Symmetric Biorthogonal Elements

In Euclidean geometry theory, orthogonality is a very important and basic concept in many fundamental theories. During studying the geometry of normed spaces, a potential subject is looking for a new concept in more general spaces to replace the concept of orthogonality in Euclidean space. With the development of the geometry of normed spaces, many scholars introduced the concept of generalized orthogonalities in general normed spaces. However, the researches about the relationships between generalized orthogonalities usually are qualitative, the differences between two generalized orthogonalities is only concerned, but how different between the different generalized orthogonalities and their influence on the geometry of spaces are rarely studied.In this paper, the existence of symmetric biorthogonal elements in classical two-dimensional normed linear spaces is mainly studied, where the biorthogonal elements are the elements which are both Birkhoff orthogonal and isosceles orthogonal, and combining with some important theories of the geometric constants which portrait the quantity difference between Birkhoff orthogonality and the isosceles orthogonality, the characterizations of biorthogonal elements are studied, then some new conclusions are obtained.First of all, some basic knowledge of the inner product space, basic knowledge and characterizations of Birkhoff orthogonality and the isosceles orthogonality are introduced, main results of previous studies are summaried, and the relevant background and significance of the results in this paper are demonstrated.Second, Birkhoff orthogonality and the isosceles orthogonality are discussed in classical normed spaces. At the same time, based on some important results in classical normed spaces, some sufficient conditions for Birkhoff orthogonality and the isosceles orthogonality are obtained. In addition, by introducing some relevant definitions we get some basic results.Finally, the existence of symmetry biorthogonal elements in some classical normed spaces is proved.