Geometric Constant Of Banach Space And Application

Geometric constant is an important tool for studying the geometric structure, so it is a hot topic to study the relationship between geometric structure and geometric constant. Recently a good deal of investigations have focuse on finding the sufficient conditions with various geometrical constants for a Banach space to have normal structure.Several modulus and constants in Banach space and their applications are studied in this thesis. Meanwhile, a new geometric constant is introduced, and the relationship among normal structure and uniform structure. The contents of this thesis are divided into three parts.First, we introduce some basic conceptions in Banach space, for example the modulus of smoothness, modulus of convexity, James constant, Jordan-von Neumann constant, and the weakly convergent sequence coefficient and so on . And we get some sufficient conditions for normal structure in Banach space.Second, we talk about the relationship between the modulus of convexity and geometric constant, A new geometric constant is introduced. We get some sufficient conditions for which a Banach space has normal structure by the relationship among the modulus of convexity, parameterized modulus of convexity and the coefficient of weak orthogonality, which generalize some well-known results of Gao Ji. Meanwhile the modulus of U -convexity and modulus of W -convexity are investigated in the samilar way, which generalize some well-known results of Gao Ji and Saejung.At last, lower bounds for the weakly convergent sequence coefficient in term of parameterized Jordan-von Neumann constant and parameterized James constant is established. Some sufficient conditions which imply normal structure are obtained by means of these bounds. Furthermore we analyse the relationship between parameterized Jordan-von Neumann constant and uniform normal structure. Our results generalize the results of Gao Ji.