Real Analysis

By Edward James McShane and Truman Arthur Botts

This text offers upper-level undergraduates and graduate students a survey of practical elements of real function theory, general topology, and functional analysis. Beginning with a brief discussion of proof and definition by mathematical induction, it freely uses these notions and techniques. The maximality principle is introduced early but used sparingly; an appendix provides a more thorough treatment. The notion of convergence is stated in basic form and presented initially in a general setting. The Lebesgue-Stieltjes integral is introduced in terms of the ideas of Daniell, measure-theoretic considerations playing only a secondary part. The final chapter, on function spaces and harmonic analysis, is deliberately accelerated. Helpful exercises appear throughout the text. 1959 edition.

• Language: English
• Publisher: Dover Publications
• Release date: Nov 7, 2013
• ISBN: 9780486165677

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INDEX

CHAPTER 0

PRELIMINARIES

§ 1. Sets. We shall constantly be concerned with sets or collections of things. If a thing x belongs to a set S, we say x is a member (or element or point) of S, and we symbolize this by writing

If x is not a member of S, we write x ∉ S. Where A and B are sets, we say A is a subset of B if every member of A is also a member of B; and we symbolize this by writing

or

If A is a subset of B, but B is not a subset of A, we say A is a proper subset of B.

We shall always use equality in the sense of logical identity: i.e., the assertion

concerning things α and β will mean simply that the symbols "α and β" are each names for the same thing. In particular for sets A and B the assertion

means that the letters A and B each name the same collection of things, i.e., that the collections A and B have exactly the same members. When things α and β are not the same, we write

and say that α and β are distinct.

It is convenient to consider that there is an empty set ∅, which by definition has no members at all. If ∅ and ∅′ are each empty sets, then ∅ ⊂ ∅′ and ∅′ ⊂ ∅, so ∅′ = ∅. Thus there is only one empty set, which we shall henceforth always denote by the symbol ∅.

The union A ∪ B of given sets A and B means the set of all things x such that x ∈ A or x ∈ B of sets means the set of all things x such that x . The intersection A ∩ B of given sets A and B means the set of all things x such that both x ∈ A and x ∈ B. If there are no such things x, then A ∩ B = ∅; and in this case the sets A and B are said to be disjointof sets means the set of all things x such that x . The (set-) difference A – B of given sets A and B means the set of all things x such that x ∈ A but x ∉ B.

A set whose members are just the things named in a certain explicit list a, b, ···, k is sometimes denoted by

In particular, the set whose only members are the things a and b may be denoted by

and the set whose only member is a may be denoted by

Evidently

The listing a, b—which in an obvious natural manner designates a as first and b as second—defines the ordered pair (a, b); if instead b is designated as first and a as second, we obtain the ordered pair (b, a).† For ordered pairs (a, b) and (c, d), we have

if and only if a = c and b = d, so that

unless a = b. The cartesian product A × B of an ordered pair (A, B) of sets means the set whose members are just those ordered pairs (a, b) for which a ∈ A and b ∈ B.

One more notational device is frequently useful. Suppose that for each thing x in a given set S, P(x) is a statement, true or false, concerning x. We shall sometimes denote the set of all members x of S for which P(x) is true by the symbol

§ 2. Functions and relations. In elementary work a function is sometimes defined as an ordered pair of things of which the first is a set, termed the domain of the function, and the second is a rule of correspondence whereby to each member x of this domain there is assigned some one thing, termed the function′s value at x. The collection of all of a given function′s values is called the range of the function. The graph of a function f is then defined to be the collection of all ordered pairs (x, f(x)) for which x belongs to the domain of f and f(x) is the value of f at x.

It may be objected that the above definition of function lacks clarity, because the notion rule of correspondence has not been carefully defined. Actually, we shall wish to think of a function as completely specified by its graph, so that two given functions will be considered to coincide if and only if they have the same graph (whether or not their rules of correspondence are identically formulated). It is thus more satisfactory simply to define a function as a graph, i.e., as a collection f of ordered pairs such that if (x, y)∈ f and (x, y′) ∈ f, then y = y′. This last stipulation assures that a function is by definition singlevalued. The domain of a function f is then the set of all first members x of pairs (x, y) belonging to the collection f, and the range of f is the set of all second members y of such pairs (x, y). When x is in the domain of a function f, we may now write f(x) for the unique y such that (x, y) ∈ f.

If the domain of a function f is a set X and the range of f is a subset of a set Y, we say that f is a function on X to Y, and we sometimes use the alternative symbol f: X → Y for f. In this case, we also sometimes use the terminology "f maps X into Y"; and in case the range of f consists of all of Y, we may say "f maps X onto Y" For a function f whose domain is the set X, we sometimes use the notation

Given a function f on a domain X, for every subset X0 of X the restriction f | X0 of f to X0 means that function on domain X0 whose value at each x in X0 is f(x).

It is an established custom, given a function f with domain X, to use the symbol f(X0) for the range of f | X0 whenever X0 is a subset of X. Thus y ∈ f(X0) if and only if y = f(x) for some x in X0. In the reverse direction, for every set E we define f –1(E) to be the set consisting of all x in X such that f(x) ∈ E; i.e.,

A function f is termed one-to-one if and only if for each y in the range Y of f there is only one x in the domain X of f such that f(x) = y—i.e., such that (x, y) ∈ f. In this case the function f has a unique inverse function f -1, consisting by definition of all pairs (y, x) for which (x, y) ∈ f. Since f is single-valued, we see that f -1 must be one-to-one. Plainly, then, given a set X and a set Y, there is a one-to-one function mapping X onto Y if and only if there is a one-to-one function mapping Y onto X. In this case X and Y are said to be cardinally equivalent, and each such function is termed a one-to-one correspondence between X and Y.

Extending the notion of function, we shall define a relation to be simply a collection of ordered pairs. Thus every function is a relation. Where ρ is a relation, we may think of

as saying "x bears relation ρ to y" or "x is ρ-related to y" and we usually express this by writing

If a relation ρ is a subset of the set S × S of all ordered pairs of elements of a set S, we say ρ is a relation in S.

Exercise

Prove that if f: X → Y is a function and A and B are sets, then

§ 3. Natural numbers and integers. The systematic study of the natural numbers 1, 2, 3, ··· and of the larger class of integers ··· –2, –1, 0, 1, 2, 3, ··· is properly part of arithmetic or number theory rather than analysis. We shall not undertake such a systematic study here. Instead, we shall assume that the reader has already acquired a thorough familiarity with the elementary properties of the natural numbers and the integers, including properties associated with their natural order, their arithmetic operations, and the use of natural numbers in counting. In this last connection, we recall specifically that a set A is termed finite if A is empty or if, for some natural number n, there is a one-to-one correspondence between A and the set {1, ···, n} of all natural numbers m for which m n. In this latter case A is said to have cardinality (or cardinal number) n. Any function f whose domain is the set {1, ···, n) is termed an ordered n-tuple, usually denoted by an expression like

where a1 = f(1), a2 = f(2), ···, an = f(n). Plainly this is in effect consistent with our earlier usage of the term ordered pair. In particular, a one-to-one function mapping the set {1, ···, n} onto itself is sometimes termed a permutation of {1, ···, n}. A set A is termed countably (or denumerably) infinite if there is a one-to-one correspondence between A and the set ω of all natural numbers. A set which is either finite or countably infinite is called countable.

For example, the set ω × ω of all ordered pairs of natural numbers is countably infinite. In fact, one way of setting up a one-to-one correspondence between this set and the set ω is suggested by the following array.

For the union ∪ (A1, ···, An} of a finite collection {A1, ···, An} of sets we may from time to time employ such variant notations as

or

or

or even, when the range of values of i is clear from context, simply

More generally, if (Aβ : β ∈ B) is a set-valued function, assigning to each in domain β a set Aβ as functional value, we sometimes use for the union ∪ {Aβ : β ∈ B} of the range {Aβ : β ∈ B} of this function such variant notations as

or simply

Analogous variant notations are used for intersections.

In 1891, G. Peano [1891] published his now famous Peano postulates for the natural number system. These essentially postulated the existence of a pair (ω, s) consisting of a set ω (called the set of natural numbers) and a function s on ω to ω (called the successor-function) for which the following three conditions hold.

P.1. There is in ω an element 1 such that for each element n of ω, 1 ≠ s(n). (In words, there is a natural number, 1, which is the successor s(n) of no natural number n.)

P.2. If m ∈ ω and n ∈ ω and s(m) = s(n), then m = n.

P.3. If ω′ is a subset of ω such that 1 ∈ ω′ and such that s(n) ∈ ω′ whenever n ∈ ω′, then ω′ = ω.

All the elementary (order- and arithmetic-) properties of the natural number system follow, as is shown in detail in, e.g., Landau [1930] or Graves [1956]. Here we merely point out that it is postulate P.3 that essentially guarantees the validity of proofs by mathematical induction. In fact, it is possible (after a short study of the resulting order relation in ω) to derive from the Peano postulates a principle of inductive proof which may be formulated in the following slightly stronger and more useful form.

(3.1) PRINCIPLE OF INDUCTIVE PROOF. Suppose for each natural number n, p(n) is a statement, true or false, concerning n. If p(1) is true, and if p(n + 1) is true whenever p(1), ···, p(n) are true, then p(n) is true for all natural numbers n.

As an application of the principle of inductive proof we can establish that the natural numbers are well-ordered by their natural order, in the sense that the following theorem holds.

(3.2) THEOREM. Every nonempty set N of natural numbers has a first element n, meaning that n ∈ N but m ∉ N for every natural number m for which m < n.

PROOF. If 1 ∈ N, this is evident. If 1 ∉ N, suppose that for every natural number n for which 1 ∉ N, ···, n ∉ N, it is the case that n + 1 ∉ N. Then by (3.1), n ∉ N for all natural numbers n, contradicting the hypothesis that N is nonempty. Therefore, there is a natural number n such that 1 ∉ N, ···, n ∉ N, but n + 1 ∈ N. That is, n + 1 is the first element of N

From the principle of inductive proof there also follows a principle of inductive definition, which asserts in effect that a uniquely defined function f = (f(n) : n = 1, 2, ···) on the set of natural numbers results from specifying f(1) and a rule whereby for each natural number n, f(n + 1) is uniquely determined by f(1), ···, f(n). We may state this in more precise terms as follows.

(3.3) THEOREM. Let Y be a nonempty set, let a be an element of Y, and let F be a function such that for every natural number n, F assigns to each ordered n-tuple (y1, ···, yn) of elements of Y a functional value F(y1, ···, yn) in Y. Then there is exactly one function f, defined on the set of natural numbers and having its values in the set Y, such that f (1) = a and such that for every natural number n,

We defer the proof of this theorem to Appendix I, contenting ourselves here with a simple illustrative example—namely, the inductive definition of the factorial function. To see how this is an instance of (3.3), let Y be the set of natural numbers, let a be 1, and let F be that function assigning to each ordered n-tuple (y1, ···, yn) of elements of Y the functional value

According to (3.3), there is exactly one function f on the set of natural numbers such that

If now for each natural number n we define n! to be f(n), we see that equations (3.4) assume the familiar form

Exercises

1. Prove that if E is a countable set, every subset of E is countable.

2. Prove that if A is a countable collection of sets, and each set belonging to A is countable, then the union of all the sets belonging to A is also countable. (Recall the discussion of ω × ω.)

§ 4. Disclaimer. This is perhaps the place to emphasize explicitly that in this book we shall attempt no systematic analysis whatever of logic or language. In particular we shall not try to pin down precisely the meanings of terms like set, statement, true, false, rule, etc. Admittedly, this leaves the precise range of applicability of principles like (3.1) undelineated. The same could be said of the notational device (1.1). Indeed, we have left entirely undiscussed the question of when a set is acceptably defined. It is sometimes said that a set is well defined provided a rule has been given whereby one can decide, for any given thing x, whether or not x belongs to the set. What is missing here, of course, is a specification of just what decision rules are to be permitted.

Questions of this character are very important for the logical foundations of mathematics and lead to surprising and deep-rooted difficulties. These have by no means been completely and satisfactorily resolved, although they are receiving intensive study today. For further information in this direction, we refer the reader to such books on mathematical logic and metamathematics as Church [1956] and Kleene [1952].

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† In the systematic study of the set-theoretic foundations of mathematics it is important to observe that the notion of ordered pair can be defined entirely in terms of sets, without reference to an intuitive comprehension of left-to-right order of listing. Essentially, the ordered pair (a, b) is specified as a two-element set

one of whose members, {a, b}, is the unordered pair involved, and the other, {a}, tells which member of this unordered pair is to be considered to be first.

CHAPTER I

REAL NUMBERS

The real number system is defined, or characterized, up to isomorphism by the requirement that it be a complete ordered field. This chapter is mainly concerned with getting at the precise meaning and justification of this statement.

§ 1. Fields. The system of real numbers is required by the above definition to form a field. A field is most smoothly defined in terms of the simpler notion of group. An (additively written) group means a pair (G, + ) consisting of a set G and a function + on G × G to G (assigning to each (x, y) in G × G a unique element x + y of G functional value) such that the following three requirements hold.

(1.1) For every x, y, and z in G, x + (y + z) = (x + y) + z.

(1.2) There is in G a unique element θ, termed the neutral or zero element, such that for every x in G,

(1.3) To each x in G there corresponds a unique inverse element –x of G such that

A group (G, +) which satisfies the following further requirement is termed commutative.

(1.4) For every x and y in G, x + y = y + x.

It follows at once from (1.3) that for each x in G, – (–x) = x. The easy verification of the following statement is left to the reader.

(1.5) Where (G, +) is a group and a and b are in G, the equation

has a unique solution x in G, namely, x = ( –a)+ b; and likewise the equation

has the unique solution y = b + (– a).

Now, a field means a triple (F, + ,·) consisting of a set F and functions + and · on F F to F such that the following three requirements are satisfied.

(1.6) The pair (F, +) is a commutative group (with zero element denoted by θ).

(1.7) For all x, y in F, x · y = y · x; and the pair (F – {θ}, ·), consisting of the set F – {θ} of non-zero elements of F together with (the restriction to domain (F – {θ(F – {θ}) of) the function ·, is a commutative group (this time multiplicatively written).

(1.8) For every x, y, and z in F, x ·(y + z) = x · y + x · z.†

In the multiplicative group (F – {θ}, ·) of a field (F, + , ·), the unique neutral, or identity, element will usually be denoted by e, and the unique inverse of an element x will usually be denoted by x–1. Notationally, we shall frequently abbreviate x·y to xy, and (F, + , ·) to F.

Given a field F, it is an easy exercise, which we again leave to the reader, to deduce that for every x in F

and, conversely, that if x and y are in F and x·y = θ, then either x = θ or y = θ (or both). This and the rephrasing of (1.5) for the multiplicative group of a field yield the following statement.

(1.9) If F is afield and a and b are in F and a ≠ θ, then the equation

has a unique solution x in F, namely, x = a–1b.

The reader may also verify without difficulty that if F is a field and x and y are in F, then

and

A subfield of a given field F means a subset H of F which, together with the functions + and · for F (as restricted to domain H H), forms a field.

(1.10) THEOREM. Let F be afield and let H be a subset of F. Then H is a subfield of F if and only if the following three conditions hold.

(i) As functions on domain H H, + and · have their ranges in H. (We sometimes express this by saying H is closed under + and ·.)

(ii) The elements θ and e of F belong to H.

(iii) If x ∈ H, then –x ∈ H; and if x ∈ H and x ≠ θ, then x–1 ∈ H.

PROOF. First, suppose H is a subfield of F. Then (i) is obvious. Now there is in H a unique zero element

θ

, which can only be θ, since

Likewise, there is in H a unique identity element

e

, which an analogous computation shows can only be e. If x ∈ H, there is in H a unique element z such that x + z =

θ

= θ. Since –x is the only element of F such that x + (– x) = θ, z must be –x. Similarly, we argue that if x ∈ H and x ≠ θ, then x–1 ∈ H.

Conversely, suppose now that (i), (ii), and (iii) hold. Then (ii) and (iii) imply that (1.2) and (1.3) hold (with H replacing G), and (1.1) and (1.4) are inherited by H from F, so (H, + ) is a commutative group. Likewise, so is (H – {θ},·). Since H inherits property (1.8) from F, we see that H is a field, and hence a subfield of F.

The following statement readily follows from (1.10), as we leave it to the reader to verify.

(1.11) The intersection of any nonempty collection of subfields of a given field F is itself a subfield of F; in particular, there is a unique "smallest" subfield of F, namely, the intersection of all subfields of F.

In the third section we shall define a notion of ordering for fields. It will turn out that every orderable field must have infinitely many elements. However, there exist fields having only finitely many elements, as the following exercise shows.

Exercise

Verify that the two-element set F = {0, 1}, with + and · defined by the following tables, is a field.

Similarly, show that F = {0, 1, 2} with + and · defined as follows is a field.

It is a fact that for any prime number p a field F = {0, 1, ···, p – 1} of exactly p elements may be defined similarly. See, e.g., McCoy [1948], p. 21.

§ 2. Associativity, commutativity, distributivity. We now turn attention to the widely useful generalized associative, commutative, and distributive laws concerning addition and multiplication in a field. Throughout this section F will denote an arbitrary field. Where n is a positive integer and (x1, ···, xn) is an ordered n-tuple of elements of F, we define the composite sum σ(x1, ···, xn) of (x1, ···, xn) inductively as follows. If n = 1, we set

If n > 1, then, supposing σ(x1, ···, xn–1) already defined, we set

We prove the following initial result, leaving most of the other proofs as exercises for the reader.

(2.1) If m and n are positive integers, and (x1, ···, xm+n) is an ordered (m + n)-tuple of elements of F, then

PROOF. The proof will be made by induction on m for fixed n. By definition

so the conclusion holds when m = 1. When m > 1, by the induction hypothesis we have

Hence,

Where n is a positive integer, consider any ordered n-tuple (x1, ···, xn) of elements of F. If n = 1, a given element u of F is called a sum of this n-tuple if and only if u = x1. Proceeding inductively, if n > 1, then an element u of F is called a sum of (x1, ···, xn) if and only if, for some positive integer m < n, there are sums v of (x1, ···, xm) and w of (xm+1, ···, xn) such that u = v + w. We note that the composite sum σ(x1, ···, xn) is a sum of (x1, ···, xn).

(2.2) If u is a sum of the ordered n-tuple (x1, ···, xn) of elements of F, then u = σ(x1, ···, xn).

The proof is left as an exercise. As a corollary we have the so-called generalized associative law for addition: for any given ordered n-tuple (x1, ···, xn) of elements of F, all of its sums coincide. We agree to denote their common value by x1 + ··· + xn. In an analogous manner we define composite product and product for (x1, ···, xn), and show that all products of (x1, ···, xn) have a common value, which we denote by x1 ··· xn.

The generalized commutative law for addition asserts that, where (x1, ··· , xn) is any ordered n-tuple of elements of F, and (1′, ··· , n′) is any permutation of {1, ··· , n},

This is evident when n is 1 or 2. To show it for general n, one first establishes by induction that

Now for some h′, n = h′, so xn = xh′. It follows from (2.4) that

By the induction hypothesis,

Substituting this in (2.5), we have (2.3). Similarly we prove a generalized commutative law for multiplication, which asserts, under the same hypothesis, that

Finally, we have the generalized distributive law, a preliminary form of which asserts that if x, y1, ··· ,yn are elements of F, then

This easily follows by induction from (1.8). More generally, we have that if x1, ··· , xm, y1, ··· , yn are elements of F, then

or, written more compactly in the familiar Σ notation,

In case m = 1, this is given by (2.6). The general case follows by induction on m for fixed n.

Henceforth we shall make free use of these laws without any specific mention that we are doing so.

§ 3. Ordered fields. An ordered field means a pair (F, > ) consisting of a field F and a relation > in F such that the following three conditions are satisfied.

(3.1) For every x in F, exactly one of the relations

holds. (As usual, "x < y" is to be understood as an alternative for the notation "y > x".)

(3.2) If x and y are in F and x > θ and y > θ, then x + y > θ and xy > θ.

(3.3) If x and y are in F, then x > y if and only if x – y > θ.

We note that (3.3) is consistent in the case where y = θ, and also that when x = θ, (3.3) asserts that for every y in F, y < θ if and only if –y > θ. We leave as exercises the proofs of the following further elementary properties of any given ordered field F.

(3.4) For every x in F, x > θ, x = θ, or x < θ according as –x < θ, –x = θ, or –x > θ.

(3.5) If x, y, z are in F and x > y, then x + z > y + z.

(3.6) For every x and y in F, exactly one of

holds.

(3.7) If x, y, z are in F and x > y and y > z, then x > z.

(3.8) For every x and y in F, if x > θ and y < θ, then xy < θ; and if x < θ and y < θ, then xy > θ.

It follows from (3.1), (3.2), and (3.8) that if x ≠ θ, then xx > θ. In particular, e = ee > θ.

(3.9) If x, y, z are in F and x < y and z > θ, then zx < zy.

(3.10) If x and y are in F and θ < x < y, then

For every x in an ordered field F, the absolute value |x| of x is defined to be x if x θ and –x if x < θ. It follows from (3.4) that for any x in F

equality holding if and only if x = θ. Where x² means xx as usual, it is easily seen that for any x in F

(In fact, since xθ, |x²| = x²| and so if x θ, |x²| = x² = |x|², whereas if x < θ, |x²| = x² = (–x)² = |x|².) We leave it to the reader to show that for every x and y in F,

Using this fact we can establish the following useful theorem.

(3.11) THEOREM. If F is an ordered field and x and y are in F, then

More generally, if x1, ··· , xn are in F, then

PROOF. We establish only the first assertion; the second follows easily by induction. We have

Therefore, since xy |xy| = |x| |y|,

It follows that |x + y|x| + |y|; for otherwise, |x + y| > |x| + |y0, whence by (3.9), |x + y|² > |x + y|(|x| + |y(|x| + |y

The order relation > in any given ordered field F evidently orders every subfield H of F, in the sense that H together with (the intersection with H H of) the relation > forms an ordered field (H, >). In particular, the smallest subfield of F is again an ordered field, termed (for reasons to appear shortly) the rational subfield of F. In order to investigate the form of rational subfields, it is convenient to introduce some notation. Where x is any element of field F and n is any positive integer, we shall write

for the sum of n terms each equal to x; and we shall define (–n)x to be –(nx). We note in passing that, since e > θ, it follows inductively from (3.2) that for every positive integer n,

From this it follows that the elements e, 2e, 3e, ··· of F are all distinct, so that an ordered field F necessarily has infinitely many elements. (If for some distinct positive integers m and n (for which, say, m > n) we have me = ne, then we see that (m – n)e = θ, contradicting (3.12).) By adding the equation θ = x + ( –x) to itself n times and rearranging slightly, we see that n(–x)= –(nx) = ( –n)x. Setting 0x equal to θ, we have now defined nx for all integers n (positive, negative, and zero) and all x in F. We leave it to the reader to verify † that, where m and n are integers and x is in F,

and, where m is an integer and x and y are in F,

Next, where m and n are integers and n ≠ 0 (and e denotes the identity in F, as usual), we define me/ne to be the unique solution of the equation

(That ne in this equation is not θ follows from (3.12).) Multiplying both sides of the equation

by any non-zero integer k and using (3.13) and (3.14),