Ebook: Mathematical Analysis: An Introduction

This is a textbook suitable for a year-long course in analysis at the ad­ vanced undergraduate or possibly beginning-graduate level. It is intended for students with a strong background in calculus and linear algebra, and a strong motivation to learn mathematics for its own sake. At this stage of their education, such students are generally given a course in abstract algebra, and a course in analysis, which give the fundamentals of these two areas, as mathematicians today conceive them. Mathematics is now a subject splintered into many specialties and sub­ specialties, but most of it can be placed roughly into three categories: al­ gebra, geometry, and analysis. In fact, almost all mathematics done today is a mixture of algebra, geometry and analysis, and some of the most in­ teresting results are obtained by the application of analysis to algebra, say, or geometry to analysis, in a fresh and surprising way. What then do these categories signify? Algebra is the mathematics that arises from the ancient experiences of addition and multiplication of whole numbers; it deals with the finite and discrete. Geometry is the mathematics that grows out of spatial experience; it is concerned with shape and form, and with measur­ ing, where algebra deals with counting.

---

The first semester of this course is the basic introductory course in analysis, introducing the words "compact", "complete" "connected", "continuous", "convergent", etc. Among traditional purposes of such a course is the training of a student in the conventions of pure mathematics: acquiring a feeling for what is considered a proof, and supplying literate written arguments to support mathematical propositions. The topics covered in the second semester, and the second half of this book, are differentiation (of vector-valued functions of several variables), integration, and the connection between these concepts which is displayed in the theorem of Stokes, in its general form. Also included are some beautiful applications of the theory such as Brouwer's fixed point theorem, and the Dirichlet principle for harmonic functions.

---

The first semester of this course is the basic introductory course in analysis, introducing the words "compact", "complete" "connected", "continuous", "convergent", etc. Among traditional purposes of such a course is the training of a student in the conventions of pure mathematics: acquiring a feeling for what is considered a proof, and supplying literate written arguments to support mathematical propositions. The topics covered in the second semester, and the second half of this book, are differentiation (of vector-valued functions of several variables), integration, and the connection between these concepts which is displayed in the theorem of Stokes, in its general form. Also included are some beautiful applications of the theory such as Brouwer's fixed point theorem, and the Dirichlet principle for harmonic functions.
Content:
Front Matter....Pages i-xiv
Real Numbers....Pages 1-27
Sequences and Series....Pages 28-54
Continuous Functions on Intervals....Pages 55-73
Differentiation....Pages 74-97
The Riemann Integral....Pages 98-122
Topology....Pages 123-154
Function Spaces....Pages 155-174
Differentiable Maps....Pages 175-200
Measures....Pages 201-222
Integration....Pages 223-252
Manifolds....Pages 253-268
Multilinear Algebra....Pages 269-284
Differential Forms....Pages 285-296
Integration on Manifolds....Pages 297-321
Back Matter....Pages 323-335

---

The first semester of this course is the basic introductory course in analysis, introducing the words "compact", "complete" "connected", "continuous", "convergent", etc. Among traditional purposes of such a course is the training of a student in the conventions of pure mathematics: acquiring a feeling for what is considered a proof, and supplying literate written arguments to support mathematical propositions. The topics covered in the second semester, and the second half of this book, are differentiation (of vector-valued functions of several variables), integration, and the connection between these concepts which is displayed in the theorem of Stokes, in its general form. Also included are some beautiful applications of the theory such as Brouwer's fixed point theorem, and the Dirichlet principle for harmonic functions.
Content:
Front Matter....Pages i-xiv
Real Numbers....Pages 1-27
Sequences and Series....Pages 28-54
Continuous Functions on Intervals....Pages 55-73
Differentiation....Pages 74-97
The Riemann Integral....Pages 98-122
Topology....Pages 123-154
Function Spaces....Pages 155-174
Differentiable Maps....Pages 175-200
Measures....Pages 201-222
Integration....Pages 223-252
Manifolds....Pages 253-268
Multilinear Algebra....Pages 269-284
Differential Forms....Pages 285-296
Integration on Manifolds....Pages 297-321
Back Matter....Pages 323-335
....