Local duality for Banach spaces

A local dual of a Banach space XX is a subspace of the dual X∗X∗ which can replace the whole dual space when dealing with finite dimensional subspaces. This notion arose as a development of the principle of local reflexivity, and it is very useful when a description of X∗X∗ is not available.We give an exposition of the theory of local duality for Banach spaces, including the main properties, examples and applications, and comparing the notion of local dual with some other weaker properties of the subspaces of the dual of a Banach space.