Introduction and Motivation: The Outstanding Story of Grothendieck’s Theorem

This chapter serves as an introduction to the main topic of the monograph. In addition to providing the basic notation and terminology, we give an overview of the historical perspective and highlight the key role and strong implications of A. Grothendieck’s seminal result, which lead to the introduction of a general theory of operator ideals (in the sense of A. Pietsch), an expansion of the “local” theory of Banach spaces, and which enabled a further discovery of deep connections between operator ideals, finite-dimensional subspaces of Banach spaces and various tensor products and norms in functional analysis.