Differentiability in Banach Spaces

There are three chapters in this work of which the first two contain differentiability results for continuous convex functions on Banach spaces. The final chapter contains differentiability results for Lipschitz isomorphisms of ℓ2. The aim of chapter 1 is to improve on a result of I. Ekeland and G. Lebourg [EL] who show that a Banach space E that admits a Lipschitz Fréchet smooth bump function is an Asplund space. It is shown that if E admits a continuous lower Fréchet smooth bump function then E is an Asplund space. Chapter 2 contains partial results towards showing that there are Gâteaux differentiability spaces that are not weak Asplund spaces. Suppose that K is a totally ordered first countable Hausdorff compact space. A topology Tw is defined on C(K) called the wedge topology, and it is shown that if every subdifferential of a continuous convex function f on C(K) contains a measure of finite support then f is Gâteaux differentiable on a τw residual set. Chapter 3 contains three examples of Lipschitz isomorphisms of ℓ2 to itself for which the derivative fails to be surjective; in the first example the Gâteaux derivative is not surjective at one point, in the second example the weak limit of limt→0(f(th) -(0))/t is zero for all h ∈ ℓ2, and in the third example the Gateaux derivative is not surjective at all points of the cube {x ∈ ℓ2 : |xi| < 2=i for all i} which is mapped affinely into a hyperplane