Real Analysis/Foreword

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Real Analysis Foreword

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Foreword

What is Analysis?

Prof. Elliot Lieb of Princeton University defines analysis as the "art of taking limits", and further adds that "estimates are the heart and soul of analysis". Mathematics is often roughly subdivided into analysis, algebra and topology, so the coverage of each of these fields is quite broad. This book is concerned in particular with analysis in the context of the real numbers — there are many other fields of analysis, such as complex analysis, functional analysis and harmonic analysis. It will first develop the basic concepts needed for the idea of functions, then move on to the more analysis-based topics.

Analysis or Calculus?

Analysis is concerned with primarily the same topics as Calculus, such as limits, derivatives, and integrals, but in a mathematical way rather than in a simply practical way. Before you study analysis, you may want to study calculus; you will end up repeating much of the same material when you come back to analysis, but you will understand its practical significance. It may seem like a wasteful duplication of effort, but you will feel much more comfortable with many of the basic concepts of analysis.

On the other hand, when studying calculus you may be dismayed at the frequent statement of rules for performing various operations with little or no justification. The study of analysis puts all these on a formal basis and provides that justification.

Preliminaries

In much of analysis, arguments must be constructed very carefully, and it must be possible to make statements very precisely. To this end, it is important to be familiar with the notation of mathematical logic, in particular the 'for all' ( $\forall$ ) and 'there exists' ( $\exists$ ) notations. This notation will be used in definitions and proofs throughout the book, and it is essential to understand what these symbols mean and how they relate to each other before attempting to understand the material presented.

The core of real analysis uses very little from other areas of mathematics. Some general fundamentals that will be useful include Set Theory, especially the sections on infinite sets and cardinality. In many places, but particularly for work on Sequences, an understanding of induction and recursion (on $\mathbb {N}$) is important; however, nothing will be required that goes beyond a standard introduction to proof technique. For some of the more advanced topics, some knowledge of Topology may be helpful. Familiarity with the concepts of Calculus will make this book easier to get through, but not necessary as all relevant concepts will be defined in the text when needed.

Note also that some minimal background in Linear Algebra is assumed in the chapter on Multivariable analysis, but this is optional as no other chapter is dependent on this one.

Advice for Readers

Like all math Wikibooks, Real Analysis is encyclopedic. Unlike most print books, it has not been designed to supplement any course or instruction. As such, it contains several ideas and concepts stacked together which might baffle the uninitiated reader. We plan to have a major reorganisation of the contents and also to provide a guide to readers so as to make the best use of this book. Also a major problem with this book in particular is the lack of comprehensive exercises, especially in the later chapters.

Editors are invited to contribute towards any of these improvements.

Authors

Martín Smith-Martínez
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