Stefano Ciaci (Tartu Ülikool)
A characterization of Banach spaces containing ℓ1(κ) via ball-covering properties
Abstract. In 1989, G. Godefroy proved that a Banach space contains an isomorphic copy of ℓ1 if and only if it can be equivalently renormed to be octahedral. It is known that octahedral norms can be characterized by means of covering the unit sphere by a finite number of balls. This observation allows us to connect the theory of octahedral norms with ball-covering properties of Banach spaces introduced by L. Cheng in 2006. Following this idea, we extend G. Godefroy’s result to higher cardinalities. We prove that, for an infinite cardinal κ, a Banach space X contains an isomorphic copy of ℓ1(κ+) if and only if it can be equivalently renormed in such a way that its unit sphere cannot be covered by κ many open balls not containing αBX, where α ∈ (0,1). We also investigate the relation between ball-coverings of the unit sphere and octahedral norms in the setting of higher cardinalities.
4. märtsil kell 10:00. Liitu Zoomis.
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