The present paper is an approach to the local theory of Banach spaces via the ultrapower construction. It includes a detailed study of ultrapowers and their dual spaces as well as a definition of a new notion, the notion of a «-extension of a Banach space. All these tools are used to give a unified definition of many classes of Banach spaces characterized by local properties (such as the tp -spaces). Many examples are given; also, as an application, it is proved that any tp -space, 1 < p < oo,

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... as an ultrapower which is isomorphic to an L^-space. [15].