Some Convexities And Smoothnesses In Banach Space And Their Applications

In this thesis, we study some convexities and smoothness in Ba-nach space and their application. Main content is as follows:In the second chapter, using four new convexities of Banach space, strong convexity, very convexity, nearly strong convexity and nearly very convexity, we give four continuous theorems for metric projection under four convexities in a nonreflective Banach space, respectively. Under the that Banach space X have some cpnvexities, we investigate the relationship among proximinal set approxinative compact set and the continuity of metric projection, we also give the representation of a metric projection P_Σ(x,α) on a hyperplane classΣ(x,α) = {f∈X^*:f(x)=α} of a dual space and prove that ifΣ(?)X^* is compact set (weakly compact), then metric projection is norm-norm (norm-weakly) upper semi-continuous.In the third chapter, the relationship between some smoothness and weak^* asymptotic-norming properities of the dual Banach space X^*. We prove that Banach space X is Frechet differentiable if and only if its dual space X^* has B(X)-ANP-I under the assumption of the X is weakly sequential complete. A new locally asymptotic-norming propertt is also introduced, and the relationship among this one and other locally asymptotic-norming properties and some topological and geometrical properties are discussed.The relationship between convexities and smoothness of Banach space is a important question in geometrical theory in Banach space. In the forth chapter, the dual properties of some convexities and smoothness are discussed. We prove necessary and sufficient conditionsof the some convexities and smoothnesses in nonreflective Banach space. The results show completely the dual relations of such convexitiesand smoothnesses and generalize relevant results about such dual problem. We also prove that strong convex (resp. very convex) is eaualivalent to almost locally uniformly rotuned (resp. weakly almost locally uniformly rotuned).β-normed space is a class pre-normed linear space and it is also a generalization of normed space. It has importmant application in quasi-subadditive functional theory. There has been little research on this. In the forth chapter, we give four general results on linear extension of isometeies between the unit spheres ofβ-normed spaces. These results improve relevant theorem inβ-normed spaced.Asplund spaces is a class important Banach space. It has many good properties and application. In the fifth chapter, an example of Asplund sapce is given, which isn't very smooth and reflective, and whose dual space hasn't property (**). This shows that Banach space X, which is only the sufficient condition for that X is Asplund space, is very smooth ( or X is reflexive or X^* has property (**)).