User:TakuyaMurata/Intro to Analysis

Preface
The book is written to provide materials that are concise yet sufficient for one to continue his study in mathematical analysis. The target audience is a senior undergraduates and beginning graduate students. Readers are expected to be principally interested in mathematical studies, and thus the materials in the book are oriented toward to theories, unless applications or concrete serve to motive theories as is the case in Lebesgue theory. The book being prepared as an introduction, the background in advanced calculus and elementary understanding of linear algebra should suffice to tackle the book. Also, this set of prerequisites allow the logical development rather than the concrete one that parallels the historical development.

For the above objective, the book is not meant to be comprehensive. I omit a number of topics like Fourier series, as albeit being nice applications they are not prerequisites for more advanced topics in analysis. On the other hand, I included a number of topics in functional analysis, as they are fundamental in advanced analysis.

The secondary objective of the book is accessibility. I made every effort to present materials as least intimidating as possible. The mathematical analysis, as contrary to the popular belief of many students, is as dull and abstract as any topic can get. It is my hope that the book gives, in addition to clarity, a glimpse of pleasure of mathematical studies.

Some materials have been adopted from "real analysis" at wikibooks.

Formalism
This short chapter discusses about what we call formalism; the way in which mathematical concepts are presented, especially in mathematical analysis, and more important why. The chapter was written with a hope to demystify the subject of what we call analysis to those new to the field, and accordingly anyone who interests himself in learning the mathematical subject, rather than really studying it, is advised to skip the chapter.

There are, crudely speaking, three ways to approach analysis: geometrical, topological and analytical approaches. Take the continuity as an example. In a geometric term, a continuous map can be considered as homeomorphism, a mapping of geometric figures under stretching, rotating and flipping, or deformation as we call it. For instance, we can say that a square is homeomorphic to a circle. In a topological term, a continuos map is precisely a mapping between open sets. In an analytical term, a function f is continuous when $$f(x)$$ -> $$f(x_0)$$ as x -> $$x_0$$. The sort of reasoning we just did is "formalism".

The key insight here is that definitions and results subsequence developed in means of proofs, like theorems, are made and coordinated for the sole purpose of constituting a formal theory, and the arrangement is only limited to one's preference. I, for one, prefers a constructive approach. The epsilon delta, for example, is not "the" definition for the continuity but one approach which came to be the norm today. The formalism thus has a consequence. As time goes by, the theory may be applied to cases that are not originally imagined when it was conceived, and this often leads to motive to alter the foundation so it "explains" newly-considered cases more naturally. Lebesgue theory is a good example of this; the theory that was invented and devised with desire to integrate over arbitrary sets. It then superseded more rudimentary Riemann integrals.

Set theory and topology
Theorem: Finite sets are closed under set theoretic operations (i.e., union, intersection and difference). Proof:

Definition: n-tuple is a set { $$a_1$$ { $$a_2$$ { $$a_3$$ ... } } } and is denoted by $$(a_1, a_2, ... a_n)$$. When n = 2, the tuple is said to be a pair.

Definition: A function is a set, which may be empty, of pairs (a, b) for a in A and b in B, denoted by f: A -> B. The pre-image of a function is { x : for any (x, y) in f }, and the image of a function is { f(x) : for any (x, y) in f }.

Example: If A = {1, 2, 3} and B = {2, 4, 6}, then f = { (a, b) for a in A and b in B }.

We observe that this definition does not specify how elements in the domain are mapped to the codomain. The importance of this generality is that there may not be an explicit form stating about the mapping. For example, in the preceding example, it is probably natural to define f(x) = 2x. But there can be another mapping; indeed, each permutation of B is a map. This means that when there are infinitely many elements, it possibly takes infinitely many statements to specify how the mapping is done. If f(x) = 2x, then its image is { f(1), f(2), f(3) }. If $$f(S) = |S|$$ for a finite set S, then { ({}, 0), ({3, 10}, 2), ({ 1, 2, ... 100 }, 100) } is a subset of the image of f.

A constant function is a function f(x) = c for some constant c. Its pre-image is the same as its codomain and its image is { c }.

It is customary to omit specifying the domain and the codomain of a function, when the domain and codomain are obvious from the context or one wants to give an impression that the function works in general. For example, the identity map f(x) = x, which is often detonated by id, can be defined for any set.

A function f: A -> B, if we consider f as a set of pairs, is said to be for x in A and y in B: It is immediate from definitions that a function is bijective if and only it is injective and surjective.
 * injective when |{(x, y)}| <= 1,
 * surjective when |{(x, y)}| >= 1, and
 * bijective when |{(x, y)}| = 1.

We also write { f > a } to mean a set of { f (x) > a }.

Theorem: Given a single-valued function f, (1) f is injective and if and only if f(a) = f(b) implies a = b. (2) f is surjective if and only if its inverse is injective. Proof: Suppose f is injective and x in the image. Then, |{ (a, x) in f }| = 1 since f is injective while (a, x) and (b, x) is both in f. Thus (a, x) = (b, x) and a = b.

Suppose $$f(\left{ x_1 + x_2 + ... x_n) / n \right}) <= f(x_1 + x_2)/(x_1 + x_2)$$ When a polynomial is symmetric; that is, permuting symbols is inconsequential, we say it is convex. More concisely, we may write f((1 - t)x_1 + tx_2) = (1 - t)f(x_1) + tf(x_2).

Fields and numbers
A field is a set with addition and multiplication defined with the following properties: (1) ab = ba and a + b = b + a (2) a(x + y) = ax + ay (distribution law) (3) (xy)z = x(yz) and (x + y) + z = x + (y + z) (associativity) (4) Division and difference are defined.

Theorem: A field has 0 and 1, either of which is unique.

Proof: a - a = 0 and a(1/a) = 1; thus, 0 and 1 exist. Moreover, 0 and 1 are unique since $$0_1 + 0_2 = 0_2 = 0_1$$. The case 1 is similar. Also notice (2) implies that a0 = a(b - b) = ab - ab = 0.

Theorem: There exists a field, which we call a real field and whose members are real numbers, with the following properties: Proof: The next chapter shows how to construct this real field by set theory.

Theorem: A real field is unique up to isomorphism; hence, we will talk of the real field.

Sequence and continuity
/Sequence and continuity

Linear space
/Linear space

Lebesgue theory
Suppose an interval [0, 1]. First remove some open subinterval (a, b) from [0, 1]. Then [0, 1] \ (a, b) is closed. Remove another open subinterval and repeat the process. After infinitely steps are taken, the set exhausts any interval. A curious fact is that the set we created in this way is non-empty; this is so since removing an open set from a closed set would not take away every point in the closed set. This set is a Cantor set, and an example of a perfect set where every point is a limit point.

Although a measure theory is a vast subject where the connection with the set theory is of great importance, Lebesgue measures can be constructed in an intuitive manner and provides the best way to get a good understanding of integrals.

A function that maps between collections of sets is called a set function in the same sense we speak of real functions or vector-valued functions.

The basic idea behind Lebesgue integrals is the volume of a geometric figure. Take a cube with a side of length a; we define its volume as a^3. Clearly, we can compute the volume of other figures by decomposing it to a collection of disjoin cubes, and adding up the volumes of all the cubes. Indeed, it is not necessary to use cubes for all the time; we can use a triangle, a square and any others as long as "if we know the volume of such a figure."

Now take a more general setting. A simple function is a function that.

Differential form
Theorem: A wedge product has the following properties: (1) (associativity) dx^dy = -dy^dx. (2) (anti-symmetry) dx^dy = -dy^dx. (3)

(2) is why every teacher tells you that you be careful in handling signs in differential forms.

Calculus of several variables
Definition: a function f is said to be linear if given a scalar a and vectors x and y, (1) f(x + y) = f(x) + f(y)

Since the function operators like a scalar, we write fx instead of the usual notation f(x). A linear function is convex.

Exercises

 * Use the stokes' theorem to show the Cauchy's integral formula.

Numbers
Definition: a rational number is a pair of integers with the following property: (1) (x, z) + (y, z) = (x + y, z) (2)

Complex number field
A complex number can also be constructed as a quotient ring of a real number field adjoined by a symbol i over ; that is in a usual notation $$\mathbb{C} = \mathbb{R}[i] / $$. Quite nicely, this is all we have to define, and one can show usual properties. After noting i^2 + 1 is irreducible, it follows that C is a field, as an irreducible polynomial generates a maximal ideal. Also, after observing $$z i^2 +  = z i^2 + z - z +  = z (i^2 + 1) - z +  = -z + $$, we obtain the product of a complex number:

$$(x_1 + y_1i)(x_2 + y_2i) +  = x_1 x_2 + y_1 y_2 i^2 + i(x_1 y_2 + y_1 x_2) +  = x_1 x_2 - y_1 y_2 + i(x_1 y_2 + y_1 x_2)$$

Here manipulations are done as if the equation is a polynomial in i.