Designed for a first course in real variables, this text presents the fundamentals for more advanced mathematical work, particularly in the areas of complex variables, measure theory, differential equations, functional analysis, and probability. Geared toward advanced undergraduate and graduate students of mathematics, it is also appropriate for students of engineering, physics, and economics who seek an understanding of real analysis. The author encourages an intuitive approach to problem solving and offers concrete examples, diagrams, and geometric or physical interpretations of results. Detailed solutions to the problems appear within the text, making this volume ideal for independent study. Topics include metric spaces, Euclidean spaces and their basic topological properties, sequences and series of real numbers, continuous functions, differentiation, Riemann-Stieltjes integration, and uniform convergence and applications.
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Yes, you can access Real Variables with Basic Metric Space Topology by Robert B. Ash in PDF and/or ePUB format, as well as other popular books in Mathematics & Topology. We have over one million books available in our catalogue for you to explore.
In a course in real analysis, the normal procedure is to begin with a definition of the real numbers, either by means of a set of axioms or by a constructive procedure which starts with the “God–given” set of positive integers. The set of all integers is constructed, and from this the rational numbers are obtained, and finally the reals. A discussion of this type is part of the area of logic and foundations rather than real analysis, and we will postpone it until much later. For now we’ll take the point of view that we know what the real numbers are: A real number is an integer plus an infinite decimal, for example, 65.7204…
If the decimal ends in all nines, we have two representations of the same real number, for example,
We will often talk about sets of real numbers, and therefore a modest amount of set–theoretic terminology is necessary before we can get anywhere. You have probably seen most of this in another course, so we will proceed rather quickly.
1.1.1Definitions and Comments
The union of two sets A and B, denoted by A∪B, is the set of points belonging to either A or B (or both; from now on, the word “or” always has the so–called inclusive connotation “or both” unless otherwise specified).
The intersection of two sets A and B, denoted by A∩B, is the set of points belonging to both A and B.
The complement of a set A, denoted by Ac, is the set of points not belonging to A. (Generally, we will be working in a fixed space Ω (sometimes called the universe), for example, the set of real numbers or perhaps the set of pairs of real numbers, that is, the Euclidean plane. All sets under discussion will consist of various points of Ω, and thus Ac is the set of points of Ω that do not belong to A).
Unions, intersections, and complements may be represented by pictures called Venn diagrams that are probably familiar to many readers; see Fig. 1.1.1.
Unions and intersections may be defined for more than two sets, in fact for an arbitrary collection of sets.
The union of sets A1, A2,…, denoted by A1∪A2∪ … or by
, is the set of points belonging to at least one of the A1; the intersection of A1, A2,…, denoted by A1∩Α2∩ … or by
, is the set of points belonging to all the Ai. The union of sets A1, …, An is often written as
, and the union of an infinite sequence A1, A2, … is denoted by
, with similar notation for intersection.
There are a few identities involving unions, intersections, and complements that come up occasionally. For example, the distributive law holds: for arbitrary sets A, B, C,
(the word “distributive” is used because in this formula, at least, intersection behaves like ordinary multiplication and union like addition).
Figure 1.1.1 Union, Intersec...
Table of contents
Cover
Title Page
Contents
1. Introduction
2. Some Basic Topological Properties of Rp
3. Upper and Lower Limits of Sequences of Real Numbers
