Some Properties Of CL-spaces And Almost CL-spaces

The theories of convexity of the unit ball of a Banach space are important parts of the geometry theory of Banach spaces. Many people study uniform convexity ,the extreme form of convexity of unit ball,but few concern its opposite extreme form of convexity. The purpose of this thesis is to study the properties of Banach spaces whose unit ball have this kind of structure.A real or complex Banach space is said to be a CL-space if its unit ball is the absolutely convex hull of every maximal covex subset of the unit sphere. If the unit ball is the closed absolutely convex hull of every maximal convex subset of the unit sphere, we say that the space is an almost CL-space.The first chapter mainly introduce the background and definitions of CL-spaces and almost CL-spaces, and we also prove that the Banach spaces C₀ and l₁ are CL-spaces by definition of CL-space.(Proposition 1.6,1.7)The main results are in the second chapter.We will show that if X is a CL-space and dim X ≥3,then X has non-CL-subspaces and we will show it from the fact that l₁³ and l_∞³ have non-CL-subspaces.Chapter 3 contains a discussion of properties of CL-spaces and almost CL-spaces containing C₀ and l₁.