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2022.01.05 Grzegorz Lewicki, UJ

Admissibility of Fréchet spaces

Let \(E\) be a Hausdorff topological vector space. A subset \( Z \) of \( E \) is said to be admissible if for every compact subset \( K \) of \( Z \) and for every neighbourhood \( V \) of zero in \( E \) there exists a continuous mapping \( H\colon K → Z \) such that \( \dim({\rm span} [H(K)])< \infty \) and \( x − Hx \in V \) for every \( x \in K \) . If \( Z = E \) we say that the space \( E \) is admissible. This notion plays an important role in the fixed point theory. The aim of this talk is to show the admissibility of some classes of Frechet spaces. As an application, it will be proved the admissibility of a large class modular spaces equipped with F-norms and norms which are certain generalizations of the classical Luxemburg F-norm and the classical Luxemburg and Orlicz norms in the convex case Also a linear version of admissibility (so-called metric approximation property) for order continuous symmetric spaces will be demonstrated. The talk is based on a joint work with Maciej Ciesielski, Instytut Matematyki Politechniki Poznańskiej.

Link to the talk.

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