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We propose a unifying approach to numerous approximation properties in Banach spaces studied from the 1930s up to our days. To do so, we introduce the concept of ideal topology and say that a Banach space E has the (I, J, tau)-approximation property if E-valued operators belonging to the operator ideal I can be approximated, with respect to the ideal topology tau, by operators belonging to the operator ideal J. This concept recovers many classical/recent approximation properties as particular instances and several important known results are particular cases of more general results that are valid in this general framework.

Ideal topologies and corresponding approximation properties