Abstract
In this chapter, general concepts connected with inner product spaces are presented. After introducing the axioms of an inner product space, a number of specific examples are given, including both familiar cases of Euclidean spaces, and a number of function spaces which are extensively used in the remainder of the book. The important case of a Hilbert space, when the space is complete with respect to the given norm arising from the inner product, receives special attention. Orthogonality, projections, and the characterization of an orthonormal basis of a Hilbert space are discussed and studied in some detail. The standard isomorphisms between a separable Hilbert space and either its dual space, or a suitable sequence space are presented.
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