Convexity And Best Approximation

In this dissertation , three problems are mainly considered . In the first chapter , we give a character for a metric projection on a Banach space's Chebyshev subspace to have linear representive , and discuss the linear representive of the metric projection on general closed sub-space having finite codimension . In the second chapter , we discuss the relationships between the locally asymptotic-norming properties , and obtain the equivalence of B(X~*)-LANP-κ and the C-κ; properties . Correspondingly , we define the strong C-κ properties and obtain the equivalence of B(X~*)-ANP-κ and the strong C-κ properties . As a consequence , we get an equivalent condition for reflectivity. Finally , we consider the locally asymptotic-norming properties from the points of view of renorming . In the last chapter , we use the duality mapping to discuss the relationships between some convexities and smoothnesses, and extend some existed conclusions .