## Vaclav Zizler : Boundaries and smoothness in Banach spaces

Let us call a subset C of the dual unit ball of X* a James boundary for a Banach space X if for every x in S_X there is f in C such that f(x)=1 . We will use this concept to show that for a separable X, the space X* is separable if X admits a norm ||.|| that is strongly subdifferentiable everywhere (i.e. the one sided derivative of ||.|| exists uniformly in the directions of S_X at each point of the space) or if Ext B_X* is separable. If Ext B_X* is countable, then we will show that X contains a copy of c_0 and admits a C^\infty -smooth norm. If for a separable X, the space X* is nonseparable, we will show that for every epsilon there is a convergent sequence {x_n} in S_X formed by points of Gateaux differentiability of the norm and such that || ||x_n||'-||x_m||'|| >= 1 - epsilon if n =/= m . Here ||x_n||' means the derivative of the norm at the point x_n . Several open problems in this area will be discussed.

#### 1. lecture

I will show that the norm closed convex hull of the extreme points of the dual ball is the ball itself if the set of the extreme points is norm separable or if X is separable and does not contain l_1, by Choquet's representation theorem (assumed). Then I will define the James boundary of a Banach space as a set S in the dual sphere such that for every x in X, |x|=f(x) for some f in S. I will show how to get similar results as above for boundaries by using James-Simons inequality or sup(limsup) lemma (1995 version)(assumed, done in Simons lectures). Then I will discuss examples of James boundaries and present a few open questions.

#### 2. lecture

I will define strongly subdifferentiable norms (SSD) (lim lim_{t to 0+} (|x+th|-|x|)/t) exists uniformly in |h|=1 for all x) and show Smulyan's like criterion for the SSD. Then I show that X* is separable if the norm of a separable X is SSD.

#### 3. lecture

I will show that if the boundary of X is countable (infinite), then X contains an isomorphic copy of c_0 and admits an equivalent C^infty norm.

#### 4. lecture

I will show James' weak compactness theorem for separable spaces and Rainwater theorem if Steve Simons will not do it and I will show that a set in a Banach space is weakly compact if it is pseudocompact in the weak topology.

#### 5. lecture

I will show that a weak compact set in a reflexive space is an intersection of finite unions of balls. Then I present a few open problems in the area of the interplay between the boundaries and smoothness.