On Some Characterizations Of Extendable Local Reflexivity Of Banach Space

In this paper, we introduce some properties of Banach lattice based on partial order relation. Then we discuss the connection between separability of Banach space and Schauder basis.Firstly,based on the Schauder basis, we describe the approximation property and show the equivalent condition by the set of compact operators and finite rank operators.Secondly, we introduce the definition of reflexivity of Banach space. Especially, each Banach space has the principle of local reflexivity. Furthermore, we introduce the definition of extendable local reflexivity of Banach space. The connections between extendable local reflexivity and approximation property are discussed in this paper. Also these connections can be generalized to the Banach lattice.