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\leftline{\bf Paseky 1997}
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\leftline{\bf 5 Talks by V\'aclav Zizler}
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\leftline{\bf Title:}
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\centerline{\bf Boundaries and smoothness in Banach spaces}
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{\sl Summary:} 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 $\|\cdot\|$ 
that is strongly subdifferentiable 
everywhere
(i.e. the one sided derivative of $\|\cdot\|$ exists 
uniformly in the directions of $S_X$ at 
each point of the space) or if  
$\text{ Ext }B_{X^*}$ is separable. 
If $\text{ Ext }B_{X^*}$ 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 $\varepsilon>0$ there is a convergent sequence
$\{x_n\}$ in $S_{X}$ formed by points of G\^ateaux 
differentiability of the norm and
such that $\|\,\|x_n\|'-\|x_m\|'\,\|\ge1-\varepsilon$ 
if $n\neq m$.
Here $\|x_n\|'$ means the derivative of the norm at the point $x_n$.
Several open problems in this area will be discussed.
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