Author	Browder, Andrew. author
Title	Mathematical Analysis [electronic resource] : An Introduction / by Andrew Browder
Imprint	New York, NY : Springer New York : Imprint: Springer, 1996
Connect to	http://dx.doi.org/10.1007/978-1-4612-0715-3
Descript	XIV, 335 p. online resource

This is a textbook suitable for a year-long course in analysis at the adยญ 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 subยญ specialties, but most of it can be placed roughly into three categories: alยญ gebra, geometry, and analysis. In fact, almost all mathematics done today is a mixture of algebra, geometry and analysis, and some of the most inยญ 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 measurยญ ing, where algebra deals with counting

1 Real Numbers -- 1.1 Sets, Relations, Functions -- 1.2 Numbers -- 1.3 Infinite Sets -- 1.4 Incommensurability -- 1.5 Ordered Fields -- 1.6 Functions on R -- 1.7 Intervals in R -- 1.8 Algebraic and Transcendental Numbers -- 1.9 Existence of R -- 1.10 Exercises -- 1.11 Notes -- 2 Sequences and Series -- 2.1 Sequences -- 2.2 Continued Fractions -- 2.3 Infinite Series -- 2.4 Rearrangements of Series -- 2.5 Unordered Series -- 2.6 Exercises -- 2.7 Notes -- 3 Continuous Functions on Intervals -- 3.1 Limits and Continuity -- 3.2 Two Fundamental Theorems -- 3.3 Uniform Continuity -- 3.4 Sequences of Functions -- 3.5 The Exponential function -- 3.6 Trigonometric Functions -- 3.7 Exercises -- 3.8 Notes -- 4 Differentiation -- 4.1 Derivatives -- 4.2 Derivatives of Some Elementary Functions -- 4.3 Convex Functions -- 4.4 The Differential Calculus -- 4.5 Lโ{128}{153}Hospitalโ{128}{153}s Rule -- 4.6 Higher Order Derivatives -- 4.7 Analytic Functions -- 4.8 Exercises -- 4.9 Notes -- 5 The Riemann Integral -- 5.1 Riemann Sums -- 5.2 Existence Results -- 5.3 Properties of the Integral -- 5.4 Fundamental Theorems of Calculus -- 5.5 Integrating Sequences and Series -- 5.6 Improper Integrals -- 5.7 Exercises -- 5.8 Notes -- 6 Topology -- 6.1 Topological Spaces -- 6.2 Continuous Mappings -- 6.3 Metric Spaces -- 6.4 Constructing Topological Spaces -- 6.5 Sequences -- 6.6 Compactness -- 6.7 Connectedness -- 6.8 Exercises -- 6.9 Notes -- 7 Function Spaces -- 7.1 The Weierstrass Polynomial Approximation Theorem . . . -- 7.2 Lengths of Paths -- 7.3 Fourier Series -- 7.4 Weylโ{128}{153}s Theorem -- 7.5 Exercises -- 7.6 Notes -- 8 Differentiable Maps -- 8.1 Linear Algebra -- 8.2 Differentials -- 8.3 The Mean Value Theorem -- 8.4 Partial Derivatives -- 8.5 Inverse and Implicit Functions -- 8.6 Exercises -- 8.7 Notes -- 9 Measures -- 9.1 Additive Set Functions -- 9.2 Countable Additivity -- 9.3 Outer Measures -- 9.4 Constructing Measures -- 9.5 Metric Outer Measures -- 9.6 Measurable Sets -- 9.7 Exercises -- 9.8 Notes -- 10 Integration -- 10.1 Measurable Functions -- 10.2 Integration -- 10.3 Lebesgue and Riemann Integrals -- 10.4 Inequalities for Integrals -- 10.5 Uniqueness Theorems -- 10.6 Linear Transformations -- 10.7 Smooth Transformations -- 10.8 Multiple and Repeated Integrals -- 10.9 Exercises -- 10.10 Notes -- 11 Manifolds -- 11.1 Definitions -- 11.2 Constructing Manifolds -- 11.3 Tangent Spaces -- 11.4 Orientation -- 11.5 Exercises -- 11.6 Notes -- 12 Multilinear Algebra -- 12.1 Vectors and Tensors -- 12.2 Alternating Tensors -- 12.3 The Exterior Product -- 12.4 Change of Coordinates -- 12.5 Exercises -- 12.6 Notes -- 13 Differential Forms -- 13.1 Tensor Fields -- 13.2 The Calculus of Forms -- 13.3 Forms and Vector Fields -- 13.4 Induced Mappings -- 13.5 Closed and Exact Forms -- 13.6 Tensor Fields on Manifolds -- 13.7 Integration of Forms in Rn -- 13.8 Exercises -- 13.9 Notes -- 14 Integration on Manifolds -- 14.1 Partitions of Unity -- 14.2 Integrating k-Forms -- 14.3 The Brouwer Fixed Point Theorem -- 14.4 Integrating Functions on a Manifold -- 14.5 Vector Analysis -- 14.6 Harmonic Functions -- 14.7 Exercises -- 14.8 Notes -- References