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Tuesday, 19 May 2026 - 10:00 to 11:30 <br />
Place: IM, konírna
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Speaker: Yoël Perreau, University of Tartu, Estonia<br />
Title: Points of continuity in Lipschitz free-spaces
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Abstract <br />
<p>In this talk, I would like present a metric characterization of points of continuity among molecules in Lipschitz free-spaces. Recall that a point $x$ on the unit sphere of a Banach space $X$ is said to be a point of weak-to-norm continuity of $B_X$, or simply a \emph{point of continuity}, if the identity mapping $Id:(B_X,w)\to (B_X,\norm{\cdot})$ is continuous at $x$. Also recall that given any complete metric space $M$, the \emph{Lipschitz free-space} $\lipfree{M}$ of $M$ is a Banach space which contains an isometric copy of $M$ in a canonical way and satisfies the following linearization property: given any Lipschitz map $f:M\to\R$, there exist a unique functional $g$ on $\lipfree{M}$ such that $g\circ \delta=f$ and $\norm{g}=\Lip(f)$, where $\delta:M\to \lipfree{M}$ is the canonical isometric embedding. A \emph{molecule} is an element of $\lipfree{M}$ of the form $m_{xy}:=\frac{\delta(x)-\delta(y)}{d(x,y)}$ where $x$ and $y$ are two distinct points in $M$. Molecules act on Lipschitz functions by computing slopes, and are the simplest possible elements that one can find in the sphere of a Lipschitz free-space. I will discuss how to relate the fact that a molecule $m_{xy}$ is a point of continuity to some local compactness of the metric space $M$ around the metric segment between the points $x$ and $y$ as well as a lack of connectability between these points. If time permits, I will also explain how this characterization can be extended to finitely supported elements in $\lipfree{M}$.</p>
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For more information see the seminar web page at <br />
https://www.math.cas.cz/index.php/events/seminar/6
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Set Theory and Analysis mailing list <br />
settfa@math.cas.cz <br />
https://list.math.cas.cz/listinfo/settfa@math.cas.cz
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