Foundations of Real and Abstract Analysis The core of this book Chapters three through five presents a course on metric normed and Hilbert spaces at the senior graduate level. The motivation for each of these chapters is the generalisation of a particular attribute of the n Euclidean space R: in Chapter 3 that attribute is distance; in Chapter 4 length; and in Chapter 5 inner product. In addition to the standard topics that arguably should form part of the armoury of any graduate student in mathematics physics mathematical economics theoretical statistics . . . this part of the book contains many results and exercises that are seldom found in texts on analysis at this level. Examples of the latter are Wong’s Theorem (3.3.12) showing that the Lebesgue covering property is equivalent to the uniform continuity property and Motzkin’s result (5. 2. 2) that a nonempty closed subset of Euclidean space has the unique closest point property if and only if it is convex. The sad reality today is that perceiving them as one of the harder parts of their mathematical studies students contrive to avoid analysis courses at almost any cost in particular that of their own educational and technical deprivation. Many universities have at times capitulated to the negative demand of students for analysis courses and have seriously watered down their expectations of students in that area. As a result mathematics majors are graduating sometimes with high honours with little exposure to anything but a rudimentary course or two on real and complex analysis often without even an introduction to the Lebesgue integral. Real Analysis Analysis on the Real Line Differentiation and the Lebesgue Integral Abstract Analysis Analysis in Metric Spaces Analysis in Normed Linear Spaces Hilbert Spaces An Introduction to Functional Analysis.