A general abstract approach to approximation properties in Banach spaces

We propose a unifying approach to many approximation properties studied in the literature from the 1930s up to our days. To do so, we say that a Banach space E has the (I,J,τ)-approximation property if E-valued operators belonging to the operator ideal I can be approximated, with respect to the topology τ, by operators belonging to the operator ideal J. Restricting τ to a class of linear topologies, which we call ideal topologies, 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 abstract framework.

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