Diameter 2 properties

Abstract

The thesis consists of a preliminary part and a main part, which has been organized as follows. Chapter 1 contains an introduction, where we explain our motivation and the goal of the thesis, and present a brief overview of our starting points. In addition to this narrative summary section, we describe the notation. In chapter 2, we recall some basic definitions and initial results. The first section deals with the weak topology of a normed space and the weak* topology of its dual space. We added this section because a student with a solid first course in functional analysis may not have seen some results mentioned here. This is followed by a section where we introduce the notion of a slice. The essential concept of this master thesis is based on slices of the unit ball. In the third section, we recall the term of an extreme point and the Krein Milman theorem. The Choquet lemma is presented next, this is used in our fifth section to prove the main result in this chapter Bourgain's lemma. Chapter 3 is the main part of this thesis. We start with the definitions of the diameter 2 properties under consideration, and establish them for classical spaces l, c0, L1[0; 1], and C0(K). It is known that Banach spaces with the Daugavet property have the strong diameter 2 property. We will verify this following the main idea but modifying slightly some details to our liking. Next we study how the diameter 2 properties are preserved by projective tensor products and lp-sums of Banach spaces. A detailed proof is given to the fact that the projective tensor product X^ Y of Banach spaces X and Y has the local diameter 2 property whenever X or Y has the local diameter 2 property. It is known that the (local) diameter 2 property is stable by taking lp-sums for all 1 p 1. On the other hand, we show that, for nontrivial Banach spaces X and Y , for all 1 < p < 1, the Banach space X p Y cannot enjoy the strong diameter 2 property whether or not X and Y have it. We end this chapter by establishing the diameter 2 properties for M- ideals. In fact, if Y is a strict M-ideal in X, then both Y and X have the strong diameter 2 property. Thus, if X is an M-ideal in X**, then both X and X** have the strong diameter 2 property. Finally, we show that if Y is an M-ideal in X, then any diameter 2 property of Y is carried to X.