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Introduction to Real Analysis by SADHAN KUMAR MAPA (SK MAPA)

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About Book ” Sk Mapa Real Analysis

This book is a two-term course in the introductory of real analysis for senior or junior mathematics majors and science students who are seriously interested in mathematics. The book is specifically designed to fill the gap, remaining in the development of calculus as it is usually presented in an elementary course, And to provide the necessary background for more advanced courses in pure and applied mathematics. The standard primary or you can say elementary Calculus sequence is the only specific prerequisite for Chapters (01 to 05), which deal with real-valued functions (real–valued functions ofreal variables are the functions that have a mapping of their subset of the set R of all the real numbers into R).

However, the other analysis-oriented courses, such as an elementary course on a differential equation, also provide a useful initial experience. Chapters 6 and 7 required a working knowledge of matrices determinants and linear transformations of matrices, Usually available in the first course of linear algebra. And Chapter 8 is accessible after the completion of the first five chapters (Chapters 01 – 05). He has also included a total 295 number of completely worked out examples in this book to illustrate and clarify all the major definitions and theorems. He has emphasized careful statements of definitions and theorems and has tried to be complete and detailed in the proofs.

In the making of the transition from one variable to several variables and from real-valued functions to vector-valued functions, He has left some proofs for the students that are essentially repetitions of the earlier theorems. He believed that working through the details of straightforward generalizations or elaborations of more elementary results is very good practice for students. Set theory is a large and complicated subject in its own right. There is no time in this course to touch on any but the simplest parts of it.

Instead, we’ll just lookat a few topics from what is often called “native set theory,” many of which should already be familiar to you.We begin with a few definitions.A set is a collection of objects called elements. Usually, sets are denoted by thecapital letters A, B, . . . , Z. A set can consist of any type and number of elements.Even other sets can be elements of a set. The sets dealt with here usually have realnumbers as their elements.If a is an element of the set A, we write a ∈ A. If a is not an element of the setA, we write a ∈/ A.If all the elements of A are also elements of B, then A is a subset of B. In thiscase, we write A ⊂ B or B ⊃ A. In particular, notice that whenever A is a set, thenA ⊂ A.

Two sets A and B are equal if they have the same elements. In this case, we write A = B. It is easy to see that A = B iff A ⊂ B and B ⊂ A. Establishing that both ofthese containments are true is the most common way to show two sets are equal.If A ⊂ B and A 6= B, then A is a proper subset of B. In cases when this isimportant, it is written A $ B instead of just A ⊂ B.There are several ways to describe a set.A set can be described in words such as “P is the set of all presidents of theUnited States.” This is cumbersome for complicated sets.All the elements of the set could be listed in curly braces as S = {2, 0, a}. If theset has many elements, this is impractical, or impossible.

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