Real analysis/Cover
Introduction
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' ( ) and 'there exists' ( ) notations. Before we get started with analysis, you might want to review some other topics, including Set Theory. Of special note are the sections on infinite sets and cardinality. In many places, but particularly for work on Sequences, an understanding of induction and recursion (on ) is important. For the more advanced topics, some knowledge of Topology may be helpful. Real analysis/Authors