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\centerline{\bf Paseky 1997, 5 lectures by V. Zizler}
\medskip
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 $\ell_1$, 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.
\medskip
2. lecture: I will define strongly subdifferentiable norms (SSD)
($\lim\limits_{t\to 0^+}\frac{\|x+th\|-\|x\|}{t}$) exists uniformly
in $\|h\|=1$ for all $x$) and show \v Smulyan's like criterion
for the SSD. Then I show that $X^*$ is separable if the norm of a separable 
$X$ is SSD.
\medskip
3. 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.
\medskip
4. 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.
\medskip
5. 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.

\medskip
I will have my lectures typed in Tex 
and bring the Tex file to Paseky for the use
by
students.

\bye


