R. Smith

will speak on

Approximation of norms in Banach spaces

Time: 4:00PM
Date: Tue 6th February 2018
Location: SCN 125 [map]

Abstract: This talk follows on from one I gave in May 2017. Let $X$ be a Banach space and let $\mathbf P$ be a property of norms. We say that a norm $\|\cdot\|$ on $X$ (equivalent to the original norm) can be approximated by norms having $\mathbf P$ if, given $\varepsilon>0$, there exists another norm $|||\cdot|||$ on $X$ with $\mathbf P$, such that $\|x\| \leq |||x||| \leq (1+\varepsilon)\|x\|$ for all $x \in X$. There are a number of papers in the literature that consider the question of whether or not all(equivalent) norms on a given space can be approximated in this way.
For a number of classes of Banach spaces $X$, including $c_0(\Gamma)$(where $\Gamma$ is an arbitrary set), certain Orlicz spaces and Lorentzpredual spaces, and a class of $C(K)$ spaces (where $K$ comes from a classof compact spaces having unbounded scattered height), we show that all equivalent norms on $X$ can be approximated by $C^\infty$-smooth norms or polyhedral norms.
This is joint work with Stanimir Troyanski, University of Murcia, Spain,and Institute of Mathematics, Bulgarian Academy of Sciences

(This talk is part of the Analysis series.)

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