x Preface measure) a secondary real analysis text can be used in conjunction with this one, but it is not strictly necessary. A small number of exercises, how- ever, will require some knowledge of point-set topology or of set-theoretic concepts such as cardinals and ordinals. A large number of exercises are interspersed throughout the text, and it is intended that the reader perform a significant fraction of these exercises while going through the text. Indeed, many of the key results and examples in the subject will in fact be presented through the exercises. In my own course, I used the exercises as the basis for the examination questions, and indicated this well in advance, to encourage the students to attempt as many of the exercises as they could as preparation for the exams. The core material is contained in Chapter 1, and already comprises a full quarter’s worth of material. Section 2.1 is a much more informal section than the rest of the book, focusing on describing problem solving strategies, either specific to real analysis exercises, or more generally, applicable to a wider set of mathematical problems this section evolved from various dis- cussions with students throughout the course. The remaining three sections in Chapter 2 are optional topics, which require understanding most of the material in Chapter 1 as a prerequisite (although Section 2.3 can be read after completing Section 1.4). Notation For reasons of space, we will not be able to define every single mathematical term that we use in this book. If a term is italicised for reasons other than emphasis or for definition, then it denotes a standard mathematical object, result, or concept, which can be easily looked up in any number of references. (In the blog version of the book, many of these terms were linked to their Wikipedia pages, or other on-line reference pages.) Given a subset E of a space X, the indicator function 1E : X R is defined by setting 1E(x) equal to 1 for x E and equal to 0 for x E. For any natural number d, we refer to the vector space Rd := {(x1,...,xd) : x1,...,xd R} as (d-dimensional) Euclidean space. A vector (x1,...,xd) in Rd has length |(x1,...,xd)| := (x1 2 + . . . + xd)1/22 and two vectors (x1,...,xd), (y1,...,yd) have dot product (x1,...,xd) · (y1,...,yd) := x1y1 + . . . + xdyd.
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