[Settfa] Tuesday 17/12, Richard Smith

[kubis at math.cas.cz]

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 Tuesday 17th December, 10:00am 
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 Speaker:Richard Smith, University College Dublin, Ireland
 Title: A topological characterization of dual strict convexity in Asplund spaces

 Abstract  

 We say that a topological space $X$ has $(*)$ if there is a sequence $(\mathscr{U}_j)_{j=0}^\infty$ of families of open subsets of $X$, with the property that given $x,y \in X$, there exists $j\in\mathbb{N}$ such that
 
\begin{enumerate}
 
\item $\{x,y\} \cap \bigcup\mathscr{U}_j$ is non-empty, and
 
\item $\{x,y\} \cap U$ is at most a singleton for all $U \in \mathscr{U}_j$.
 
\end{enumerate}
 
This property was introduced by J.~Orihuela, S.~Troyanski and the author several years ago, in relation to strictly convex norms on Banach spaces. It is a simultaneous generalisation of the $G_\delta$-diagonal property and a topological property introduced by Gruenhage in 1987.
 

 
We show that if $X$ is an Asplund space, then it admits an equivalent norm having a strictly convex dual norm if and only if the dual unit sphere $S_{X^*}$ (equivalently $X^*$), endowed with the $w^*$-topology, possesses $(*)$. It follows that this ostensibly geometric property of the space can in fact be characterized in purely non-linear, topological terms. This improves upon a previous characterization, obtained by the authors above, which required an additional linearity assumption.

 For more information see the seminar web page at 
 https://calendar.math.cas.cz/set-theory-and-analysis-actual .