This chapter lays the mathematical foundation for the study of optimization that occupies the rest of this book. It focuses on three main topics: the topological structure of Euclidean spaces, continuous and differentiable functions on Euclidean spaces and their properties, and matrices and quadratic forms. Readers familiar with real analysis at the level of Rudin (1976) or Battle (1964), and with matrix algebra at the level of Munkres (1964) or Johnston (1984, Chapter 4), will find this chapter useful primarily as a refresher; for others, a systematic knowledge of its contents should significantly enhance understanding of the material to follow.
Since this is not a book in introductory analysis or linear algebra, the presentation in this chapter cannot be as comprehensive or as leisurely as one might desire. The results stated here have been chosen with an eye to their usefulness towards the book's main purpose, which is to develop a theory of optimization in Euclidean spaces. The selective presentation of proofs in this chapter reveals a similar bias. Proofs whose formal structure bears some resemblance to those encountered in the main body of the text are spelt out in detail; others are omitted altogether, and the reader is given the choice of either accepting the concerned results on faith or consulting the more primary sources listed alongside the result.
It would be inaccurate to say that this chapter does not presuppose any knowledge on the part of the reader, but it is true that it does not presuppose much.
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