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Mathematical Analysis

An Introduction

verfasst von: Andrew Browder

Verlag: Springer New York

Buchreihe : Undergraduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

1. Real Numbers

Abstract

In this chapter, we describe the system of real numbers, deducing some of their essential properties from the axioms for a complete ordered field. Before doing so, we take a quick look at the ideas and notations of sets, relations, and functions, sketch the construction of the integers and the rational numbers (starting from the natural numbers), and indicate the need for a field larger than the rational numbers. At the end of the chapter, we sketch the proof of the existence and (essential) uniqueness of a complete ordered field.

Andrew Browder

2. Sequences and Series

Abstract

In the first chapter, we defined a sequence in X to be a mapping from N to X. Let us broaden this definition slightly, and allow the mapping to have a domain of the form $\left\{ {n \in {\rm Z}:m\underline < n\underline < p} \right\},or\left\{ {n \in {\rm Z}:n\underline > m} \right\}$, for some m ∈ Z (usually, but not always, m = 0 or m = 1). The most common notation is to write n→ x_n instead of n→ x(n). If the domain of the sequence is the finite set $\left\{ {m,m + 1, \ldots ,p} \right\}$, we write the sequence as $\left( {x_n } \right)_{n = m}^p$, and speak of a finite sequence (though we emphasize that the sequence should be distinguished from the set $\left. {\left\{ {x_n :m\underline < n\underline < p} \right\}} \right)$. If the domain of the sequence is a set of the form $\left\{ {m,m + 1,m + 2, \ldots } \right\} = \left\{ {n \in {\rm Z}:n\underline { > m} } \right\}$, we write it as $(x_n )_{n = m}^\infty$, and speak of an infinite sequence. Note that the corresponding set of values $\left\{ {x_n :n\underline > m} \right\}$ may be finite. When the domain of the sequence is understood from the context, or is not relevant to the discussion, we write simply (x_n). In this chapter, we shall be concerned with infinite sequences in R.

Andrew Browder

3. Continuous Functions on Intervals

Abstract

In this chapter, we begin the study of continuous functions with the special case of real-valued functions defined on an interval in R. The concepts we develop here will be reexamined in a more general setting in later chapters.

Andrew Browder

4. Differentiation

Abstract

In this chapter we develop the basic theory of derivatives of real-valued functions of one variable. The geometric motivation of the derivative of f at c is that it is the limit of the slope of lines joining (c, f(c)) to nearby points on the graph of f, as these points approach c, and therefore represents the slope of a “line passing through two consecutive points” of the graph of f, i.e., the tangent line. We can enjoy such language without succumbing to it.

Andrew Browder

5. The Riemann Integral

Abstract

In this chapter we give an exposition of the definite integral of a real-valued function defined on a closed bounded interval. We assume familiarity with this concept from a previous study of calculus, but want to develop the theory in a more precise way than is typical for calculus courses, and also take a closer look at what kind of functions can be integrated. The integral to be defined and studied here is now widely known as the Riemann integral; in a later chapter we will study the more general Lebesgue integral.

Andrew Browder

6. Topology

Abstract

In this chapter, we extend the notions of neighborhoods, convergent sequences, and continuous functions, which we have studied in the setting of the real line, to more general situations. We are interested especially in the setting of R^d, the Euclidean space of dimension d, but it turns out that the means by which we formulate and analyze the notions of continuity and convergence, and related ideas, carry over to much more general settings, with little or no adaptation; as a result, we shall introduce some fairly abstract notions right from the beginning.

Andrew Browder

7. Function Spaces

Abstract

In this chapter, we give a few applications of the results obtained in the preceding chapters, especially the last chapter. The common ground is that we are considering spaces whose points are functions, and functions on such spaces.

Andrew Browder

8. Differentiable Maps

Abstract

If f is a real-valued function on an interval (a, b), we recall that f is differentiable at the point c ∈ (a, b) if

$$\mathop {\lim }\limits_{t \to c} \frac{{f(t) - f(c)}}{{t - c}}$$

exists; if it does, we denote its value by f′(c), and call it the derivative of f at c. In the last chapter, we remarked that this definition makes sense for a complex-valued function on (a,b). If f : (a,b) → Rⁿ is a vector-valued function on (a, b), the same definition still makes perfect sense. If f = (f₁,…, f_n), we see that f′ exists if and only if f′ exists for each j, 1 ≤ j ≤ n, and that in this case, f′ = (f′₁,… f′_n). If we try to extend the definition in another direction, however, we run into trouble. If f is a real-valued function defined in some neighborhood of the point c ∈ Rⁿ, the definition above makes no sense for n < 1, since we can’t divide by vectors. We are led to the right idea by focusing our attention not on the number f′(c), which as we know represents the slope of the tangent line to the graph of f, but on the tangent line itself, which is the graph of a linear function.

Andrew Browder

9. Measures

Abstract

In Chapter 5 we defined the Riemann integral of a real function f over a bounded interval [a, b] by

$$\int_a^b {f\left( x \right)} dx = \lim \sum\limits_{j = 1}^n {f\left( {\xi _j } \right)} \left( {x_j - x_{j - 1} } \right),$$

where x_j-1 ≤ ξ_j ≤ x_j for each j, and the limit is taken over increasingly fine partitions a = x₀ < x₁ < … < x_n= b of the interval. We found that this limit existed whenever f was continuous on [a,b], in fact, whenever f was bounded, with a set of discontinuities D which was “small,” in the sense that for any ε > 0, there existed a finite collection of open intervals $\left\{ {\left( {a_k ,b_k } \right):k = 1, \ldots r} \right\}$ such that

$$D \subset \mathop \cup \limits_{k = 1}^r \left( {a_k ,b_k } \right)and\sum\limits_{k = 1}^r {\left( {b_k - a_k } \right)} < \epsilon$$

This is a fairly rich class of functions, including as it does not only every continuous function, but also some functions which have infinitely many, even uncountably many, discontinuities (recall that the Cantor set is small in the above sense.) However, the class of Riemann integrable functions does have at least one glaring weakness: it is not stable under pointwise convergence. That is, if f_n is Riemann integrable for each n, and if f_n(x) → f(x) for every x, a ≤x ≤b, it is entirely possible that f is not Riemann integrable. (For instance, take a = 0 and b=1, and set f_n(x) = 1 if x = m/n! for some integer m, and f_n(x) = 0 otherwise. Then each f_n is Riemann integrable, and f_n converges pointwise to the function f, where f(x) = 1 if x is rational, and f(x) = 0 when x is irrational. We have seen that f is not Riemann integrable.

Andrew Browder

10. Integration

Abstract

We now turn to the topic of integration. While our main interest is in Lebesgue integration in Rⁿ, we develop the general theory of integration with respect to an arbitrary measure—it is not harder to do, and there will be occasion to use the extra generality.

Andrew Browder

11. Manifolds

Abstract

In this chapter we formulate the notion of a manifold, which generalizes the familiar ideas of a (smooth) curve or surface. The intuitive idea of a curve in R² or R³ is that of a subset C of R² or R³ which locally looks like a segment of the real line; in other words, if p ∈ C, there should be a neighborhood U of p in R² and an interval (a, b) in R, together with a bijective map a : (a, b) → C ∩ U which is bicontinuous. If p is an endpoint of C, the interval (a, b) should be replaced by an interval [a, b) or (a, b]. This formulation is purely topological, i.e., defined in terms of continuity; we want to restrict our attention to differentiate curves, which should mean that α and α^-1 are required to be differentiate. We have to explain what we mean by α^-1 being differentiable, since its domain is not an open subset of R³. We will also formulate the notion of the tangent space to a differentiate manifold, generalizing the familiar ideas of tangent line to a curve or tangent plane to a surface, and discuss the idea of orientation.

Andrew Browder

12. Multilinear Algebra

Abstract

Throughout this chapter, V will denote a vector space of dimension n over the reals.

Andrew Browder

13. Differential Forms

Abstract

Having studied tensors and alternating tensors from the purely algebraic point of view, we now consider functions whose values are tensors, or especially, alternating tensors. These are called tensor fields, and alternating tensor fields are called differential forms. They have many applications in geometry and analysis, as well as physics.

Andrew Browder

14. Integration on Manifolds

Abstract

In this chapter, we define the integral of a k-form over a compact oriented k-manifold, and prove the important generalized Stokes’ theorem, which can be regarded as a far-reaching generalization of the fundamental theorem of calculus. We also define the integral of a function over a (not necessarily oriented) manifold, and describe the integral of a form in terms of the integral of a function. The classical theorems of vector analysis (Green’s theorem, divergence theorem, Stokes’ theorem) appear as special cases of the general Stokes’ theorem. Applications are made to topology (the Brouwer fixed point theorem) and to the study of harmonic functions (the mean value property, the maximum principle, Liouville’s theorem, and the Dirichlet principle).

Andrew Browder

Backmatter

Titel: Mathematical Analysis
verfasst von: Andrew Browder
Verlag: Springer New York
Electronic ISBN: 978-1-4612-0715-3
Print ISBN: 978-1-4612-6879-6
DOI: https://doi.org/10.1007/978-1-4612-0715-3

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Real Numbers

2. Sequences and Series

3. Continuous Functions on Intervals

4. Differentiation

5. The Riemann Integral

6. Topology

7. Function Spaces

8. Differentiable Maps

9. Measures

10. Integration

11. Manifolds

12. Multilinear Algebra

13. Differential Forms

14. Integration on Manifolds

Backmatter