Homological dimensions of Banach spaces

2019/08/18 by Sánchez, Félix Cabello, Castillo, Jesús M . F., García, Ricardo · 1 citation
#46M10 #46M15 #46M18 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1908.06526

Abstract

The purpose of this paper is to lay the foundations for the study of the problem of when \Extⁿ(X, Y)=0 in Banach/quasi-Banach spaces. We provide a number of examples of couples X,Y so that \Extⁿ(X,Y) is (or is not ) 0, including the first example of a separable Banach space \mathscr K so that \Extⁿ(\mathscr K, \mathscr K)≠ 0 for all n∈ \N. Such space moreover provides the first example of Banach spaces with infinite homological dimension/codimension. We also show that the homological dimension/codimension of Hilbert spaces is infinite. The final section is devoted to compare \Ext²(⋅, ⋅) in Banach and Quasi-Banach spaces.

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