Advanced Calculus with Linear Analysis

Advanced Calculus with Linear Analysis provides information pertinent to the fundamental aspects of advanced calculus from the point of view of linear spaces. This book covers a variety of topics, including function spaces, infinite series, real number system, sequence spaces, power series, partial differentiation, uniform continuity, and the class of measurable sets.

Organized into nine chapters, this book begins with an overview of the concept of a single-valued function, consisting of a rule, a domain, and a range. This text then describes an infinite sequence as an ordered set of elements that can be put into a one-to-one correspondence with the positive integers. Other chapters consider a normed linear space, which is complete if and only if every Cauchy sequence converges to an element in the space. This book discusses as well the convergence of an infinite series, which is determined by the convergence of the infinite sequence of partial sums.

This book is a valuable resource for students.

Preface

Acknowledgments

Summary of Notation for Linear Spaces

List of Elementary Symbols

I. Function Spaces

1.1 Linear Spaces

1.2 Normed Spaces

1.3 Some Inequalities

II. Sequence Spaces and Infinite Series

2.1 Sequences

2.2 Sequence Spaces

2.3 Infinite Series—Tests for Convergence

2.4 Additional Convergence Tests—Alternating Series

2.5 Convergence Sets for Power Series

2.6 Operations on Series

III. Completeness Properties

3.1 Completeness of the Real Number System

3.2 Norm Convergence

3.3 Completeness of Sequence Spaces

IV. Continuous Functions

4.1 Completeness of C[a, b]—Uniform Continuity

4.2 Properties of Continuous Functions

4.3 Some Topological Concepts

4.4 Functions of Two Variables

V. Differentiable Functions

5.1 Preliminary Theorems

5.2 Completeness of C(1)[a, b]

5.3 Partial Differentiation

5.4 Taylor's Formula—Analytic Functions

VI. Riemann Integrable Functions

6.1 The Riemann Integral

6.2 Antiderivatives—Differentiation of the Integral

6.3 Improper Integrals

6.4 Convergence Problems

VII. Infinite Series of Functions

7.1 Functions Expressed as Infinite Series

7.2 Power Series

VIII. Lebesgue Measure

8.1 The Measure of a Bounded Open Set

8.2 Outer Measure—The Measure of a Bounded Set

8.3 The Class of Measurable Sets

IX. Lebesgue Integrable Functions

9.1 Measurable Functions

9.2 Lebesgue Integral of Nonnegative Measurable Functions

9.3 Lebesgue Integrable Functions—Convergence Properties

9.4 The Space 𝔏P

Appendix

Suggestions for Further Reading

Solutions to Selected Problems

Subject Index