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Über dieses Buch

Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.

New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points — which are rarely treated in books at this level — and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.

Inhaltsverzeichnis

1. Real Numbers

Before we discuss the system of real numbers it is best to make a few general remarks about mathematical outlook.

Charles C. Pugh

2. A Taste of Topology

It may seem paradoxical at first, but a specific math problem can be harder to solve than some abstract generalization of it. For instance, if you want to know how many roots the equation

t

5

4

t

4

+

t

3

t

+

1

=

0

$$\displaystyle{t^{5} - 4t^{4} + t^{3} - t + 1 = 0}$$

can have then you could use calculus and figure it out. It would take a while. But thinking more abstractly, and with less work, you could show that every

n

th

-degree polynomial has at most

n

roots. In the same way many general results about functions of a real variable are more easily grasped at an abstract level—the level of metric spaces.

Charles C. Pugh

3. Functions of a Real Variable

The function

f

:

(

a

,

b

)

$$(a,b) \rightarrow \mathbb{R}$$

is

differentiable at

differentiable function

x

if

1

lim

t

x

f

(

t

)

f

(

x

)

t

x

=

L

$$\displaystyle{ \lim _{t\rightarrow x}\frac{f(t) - f(x)} {t - x} = L }$$

exists. This means

L

is a real number and for each

ε

> 0 there exists a

δ

> 0 such that if 0 <  |

t

x

|  <

δ

then the

differential quotient

differential quotient

above differs from

L

by <

ε

. The limit

L

is the

derivative

derivative

of

f

at

x

,

L

=

f

(

x

). In calculus language,

Δ

x

=

t

x

$$\varDelta x = t - x$$

is the change in the independent variable

x

while

Δ

f

=

f

(

t

)

f

(

x

)

$$\varDelta f = f(t) - f(x)$$

is the resulting change in the dependent variable

y

=

f

(

x

). Differentiability at

x

means that

f

(

x

)

=

lim

Δ

x

0

Δ

f

Δ

x

.

$$\displaystyle{f^{{\prime}}(x) =\lim _{\varDelta x\rightarrow 0}\frac{\varDelta f} {\varDelta x}.}$$

We begin by reviewing the proofs of some standard calculus facts.

Charles C. Pugh

4. Function Spaces

Points converge to a limit if they get physically closer and closer to it.

Charles C. Pugh

5. Multivariable Calculus

This chapter presents the natural geometric theory of calculus in

n

dimensions.

Charles C. Pugh

6. Lebesgue Theory

This chapter presents a geometric theory of Lebesgue measure and integration. In calculus you certainly learned that the integral is the area under the curve. With a good definition of area that is the point of view I advance here. Deriving the basic theory of Lebesgue integration then becomes a matter of inspecting the right picture. See Appendix E for the geometric relation between Riemann integration and Lebesgue integration.

Charles C. Pugh

Backmatter

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