This chapter presents preliminary material from functional analysis that will be used in subsequent chapters. Most of the results are stated without proofs, as they are standard and can be found in many references. We start with a review of de.nitions and properties of linear normed spaces and Banach spaces, including results on duality and weak convergence. We then recall some properties of the Hilbert spaces. Finally, we present miscellaneous results that will be applied repeatedly in this book; they include elements of convex analysis, .xed point theorems, and well-known inequalities. All the linear spaces considered in this book including abstract normed spaces, Banach spaces, Hilbert spaces, and various function spaces are assumed to be real spaces. We assume that the reader has some knowledge of linear algebra and general topology.
KeywordsHilbert Space Banach Space Linear Space Normed Space Weak Convergence
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