Real Analysis

2

Real Analysis

Travis Dirle

December 4, 2016

2

Contents

1 Families of Sets 1

2 Measures 3

3 Construction of Measures 5

4 Measurable Functions 7

5 The Lebesgue Integral 9

6 Limit Theorems 11

7 Differentiation and Integration 13

8 Product Measures 17

9 Signed Measures 19

10 Lp Spaces 21

11 The Radon-Nikodym Theorem 25

12 Linear Operators 27

13 Hilbert Spaces 31

i

CONTENTS

ii

Chapter 1

Families of Sets

Definition 1.0.1. An algebra is a collection A of subsets of X such that(i) ∅ ∈ A and X ∈ A;(ii) if A ∈ A then Ac ∈ A;(iii) if A1, . . . , An ∈ A, then ∪ni=1Ai and ∩ni=1Ai are in A.

A is a σ-algebra if in addition(iv) whenever A1, A2, . . . are in A, then ∪∞i=1Ai and ∩∞i=1Ai are in A.

Since ∩∞i=1Ai = (∪∞i=1Aci)c, the requirement that ∩∞i=1Ai be inA is redundant.

The pair (X,A) is called a measurable space. A set A is measurable or Ameasurable if A ∈ A.

Lemma 1.0.2. If Aα is a σ-algebra for each σ in some non-empty index set I,then ∩α∈IAα is a σ-algebra.

If G is the collection of open supsets of X, then we call σ(G) the Borel σ-algebra on X, and this is often denoted B. Elements of B are called Borel setsand are said to be Borel measurable. When X = R, B is not equal to 2R.

Proposition 1.0.3. If X = R, then the Borel σ-algebra B is generated by eachof the following collection of sets:

(i) C1 = {(a, b) : a, b ∈ R};(ii) C2 = {[a, b] : a, b ∈ R};(iii) C3 = {(a, b] : a, b ∈ R};(iv) C4 = {(a,∞) : a ∈ R}:

1

CHAPTER 1. FAMILIES OF SETS

2

Chapter 2

Measures

Definition 2.0.1. Let X be a set and A a σ-algebra consisting of subsets of X. Ameasure on (X,A) is a function µ : A → [0,∞] such that

(i) µ(∅) = 0;(ii) if Ai ∈ A, i = 1, 2, . . . , are pairwise disjoint, then

µ(∪∞i=1Ai) =∑∞

i=1 µ(Ai).

The second condition is known as countable additivity. We say a set func-tion is finitely additive if µ(∪ni=1Ai) =

∑ni=1 µ(Ai) whenever A1, . . . , An are in

A and are pairwise disjoint. The triple (X,A, µ) is called a measure space.

Proposition 2.0.2. The following hold:(i) If A,B ∈ A with A ⊂ B, then µ(A) ≤ µ(B).(ii) If Ai ∈ A and A = ∪∞i=1Ai, then µ(A) ≤

∑∞i=1 µ(Ai).

(iii) Suppose Ai ∈ A and Ai ↑ A. Then µ(A) = limn→∞ µ(An).(iv) Suppose Ai ∈ A and Ai ↓ A. If µ(A1) < ∞, then we have µ(A) =

limn→∞ µ(An).

Definition 2.0.3. A measure µ is a finite measure if µ(X) < ∞. A measure µis σ-finite if there exist sets Ei ∈ A for i = 1, 2, . . . such that µ(Ei) < ∞ foreach i and X = ∪∞i=1Ei. If µ is a finite measure, then (X,A, µ) is called a finitemeasure space, and similarly, if µ is a σ-finite measure, then (X,A, µ) is calleda σ-finite measure space.

Let (X,A, µ) be a measure space. A subset A ⊂ X is a null set if thereexists a set B ∈ A with A ⊂ B and µ(B) = 0. We do not require A to be in A.If A contains all the null sets, then (X,A, µ) is said to be a complete measurespace. The completion of A is the smallest σ-algebra A containing A such that(X,A, µ) is complete.

3

CHAPTER 2. MEASURES

4

Chapter 3

Construction of Measures

Definition 3.0.1. Let X be a set. An outer measure is a function µ∗ defined on2X satisfying

(i) µ∗(∅) = 0;(ii) if A ⊂ B, then µ∗(A) ≤ µ∗(B);(iii) µ∗(∪∞i=1Ai) ≤

∑∞i=1 µ

∗(Ai) whenever A1, A2, . . . are subsets of X.

A set N is a Null set with respect to µ∗ if µ∗(N) = 0. A common way togenerate outer measures is as follows.

Proposition 3.0.2. Suppose C is a collection of subsets of X such that ∅ and Xare both in C. Suppose ` : C → [0,∞] with `(∅) = 0. Define

µ∗(E) = inf{∞∑i=1

`(Ai) : Ai ∈ C for each i and E ⊂ ∪∞i=1Ai}.

Then µ∗ is an outer measure.

We wll see that while µ∗ is an outer measure, it is not a measure on 2R.We will also see that if we restrict µ∗ to a σ-algebra L which is strictly smallerthan 2R, then µ∗ will be a measure on L. In the above proposition, when C isthe collection of intervals, and ` is merely the length of an interval, then thatmeasure is what is known as Lebesque measure. The σ-algebra L is called theLebesque σ-algebra.

Definition 3.0.3. Let µ∗ be an outer measure. A set A ⊂ X is µ∗-measurable if

µ∗(E) = µ∗(E ∩ A) + µ∗(E ∩ Ac)

for all E ⊂ X .

Theorem 3.0.4. (Caratheodory Theorem) If µ∗ is an outer measure on X, thenthe collection A of µ∗-measurable sets is a σ-algebra. If µ is the restriction ofµ∗ to A, then µ is a measure. Moreover, A contains all the null sets.

From here on, we shall usually use ’m’ instead of µwhen talking of Lebesquemeasures

5

CHAPTER 3. CONSTRUCTION OF MEASURES

Proposition 3.0.5. Every set in the Borel σ-algebra on R is m∗-measurable.

Proposition 3.0.6. If e and f are finite and I = (e, f ], then m∗(I) = `(I).

In the special case of Lebesgue measure, the collection ofm∗-measurable setsis called the Lebesgue σ-algebra. A set is Lebesgue measurable if it is in theLebesgue σ-algebra.

The Lebesgue measure of a coutable set is 0. However, there are uncountablesets which have Lebesgue measure 0, like the Cantor set.

The countable intersections of open sets are sometimes called Gδ sets; Thecountable unions of closed sets are called Fσ.

Theorem 3.0.7. Let E be any set of real numbers. Then each of the followingassertions is equivalent to the measurablilty of E.

(i) For each ε > 0, there is an open set O containing E for which m∗(O −E) < ε.

(ii) There is a Gδ set G containing E for which m∗(G− E) = 0.(iii) For each ε > 0, there is a closed set F contained in E for which m∗(E −

F ) < ε.(iv) There is an Fσ set F contained in E for which m∗(E − F ) = 0.

A measure on an algebra is called a premeasure.

Theorem 3.0.8. (Caratheodory Extension Theorem) Suppose A0 is an algebraand ` : A0 → [0,∞] is a premeasure on A0. Define

µ∗(E) = inf{∑∞

i=1 `(Ai) : each Ai ∈ A0, E ⊂ ∪∞i=1Ai}

for E ⊂ X . Then(i) µ∗ is an outer measure;(ii) µ∗(A) = `(A) if A ∈ A0;(iii) every set in A0 is µ∗-measurable;(iv) if ` is σ-finite, then there is a unique extension to σ(A0).

6

Chapter 4

Measurable Functions

Definition 4.0.1. A function f : X → R is measurable or A measurable if{x : f(x) > a} ∈ A for all a ∈ R.

Proposition 4.0.2. Suppose f is real-valued. The following conditions are equiv-alent:

(i) {x : f(x) > a} ∈ A for all a ∈ R;(ii) {x : f(x) ≤ a} ∈ A for all a ∈ R;(iii) {x : f(x) < a} ∈ A for all a ∈ R;(iv) {x : f(x) ≥ a} ∈ A for all a ∈ R;

Proposition 4.0.3. If X is a metric space, A contains all the open sets, andf : X → R is continuous, then f is mearsurable.

Proposition 4.0.4. Let c ∈ R. If f and g are measurable real-valued functions,then so are f + g,−f, cf, fg, max(f, g), and min(f, g).

Proposition 4.0.5. If fi is a measurable real-valued function for each i, then soare supifi, infifi, lim supi→∞fi, and lim infi→∞fi.

Definition 4.0.6. We say f = g almost everywhere, written f = g a.e., if{x : f(x) 6= g(x)} has measure zero. Similarly, we say fi → f a.e. if teh set ofx where fi(x) does not converge to f(x) has measure zero.

If f : R → R is measurable with respect to the Lebesgue σ-algebra, we sayf is Lebesgue measurable.

Proposition 4.0.7. If f : R→ R is monotone, then f is Borel measurable henceLebesgue measurable.

Proposition 4.0.8. Let (X,A) be a measurable space and let f : X → R be anA measurable function. If A is in the Borel σ-algebra on R, then f−1(A) ∈ A.

Definition 4.0.9. Let (X,A) be a measurable space. If E ∈ A, define thecharacteristic function of E by

χE(x) =

{1, x ∈ E;0, x 6∈ E.

7

CHAPTER 4. MEASURABLE FUNCTIONS

A simple function s is a function of the form

s(x) =n∑i=1

aiχEi(x)

for real numbers ai and measurable sets Ei.

Proposition 4.0.10. Suppose f is a non-negative and measurable function. Thenthere exists a sequence of non-negative measurable simple functions sn increas-ing to f.

Theorem 4.0.11. (Lusin’s Theorem) Let f be a real-valued measurable functionon E. Then for each ε > 0, there is a continuous function g on R and a closedset F contained in E for which

f = g on F and m(E − F ) < ε.

Theorem 4.0.12. (Egoroff’s Theorem) Assume E has finite measure. Let {fn}be a sequence of measurable functions on E that converge pointwise on E to thereal-valued function f. Then for each ε > 0, there is a closed set F contained inE for which

{fn} → f uniformly on F and m(E − F ) < ε.

8

Chapter 5

The Lebesgue Integral

Definition 5.0.1. Let (X,A, µ) be a measure space. If

s =n∑i=1

aiχEi

is a non-negative measurable simple function, define the Lebesgue integral of sto be ∫

s dµ =n∑i=1

aiµ(Ei).

If f ≥ 0 is a measurable function, define∫f dµ = sup

{∫s dµ : 0 ≤ s ≤ f, s simple

}.

Let f be measurable and let f+ = max(f, 0) and f− = max(−f, 0). So f =f+ − f− and |f | = f+ + f−. Provided

∫f+ dµ and

∫f− dµ are not both

infinite, define ∫f dµ =

∫f+ dµ−

∫f− dµ.

Definition 5.0.2. If f is measurable and∫|f | dµ <∞, we say f is integrable.

Theorem 5.0.3. (Linearity and Monotonicity of Integration) Let the functionsf and g be integrable over E. Then for any a and b, the function af + bg isintegrable over E and ∫

E

(af + bg) = a

∫E

f + b

∫E

g.

Moreover,

if f ≤ g on E, then∫E

f ≤∫E

g.

The integral∫fχA dµ is often written

∫Af dµ.

9

CHAPTER 5. THE LEBESGUE INTEGRAL

Corollary 5.0.4. (Additivity Over Domains of Integration) Let f be integrableover E. Assume A and B are disjoint measurable subsets of E. Then∫

A∪Bf =

∫A

f +

∫B

f.

Theorem 5.0.5. (The Countable Additivity of Integration) Let f be integrableover E and {En}∞n=1 a disjoint countable collection of measurable subsets of Ewhose union is E. Then ∫

E

f =∞∑n=1

∫En

f.

Theorem 5.0.6. (The Continuity of Integration) Let f be integrable over F.(i) If En ↑ E is a countable collection of measurable subsets of F, then∫

E

f = limn→∞

∫En

f

(ii) If En ↓ E is a descending countable collection of measurable subsets ofF, then ∫

E

f = limn→∞

∫En

f.

Theorem 5.0.7. Let f be a bounded measurable function on a set of finite mea-sure E. Then f is integrable over E.

Proposition 5.0.8. If f is integrable,∣∣∣∣∫ f

∣∣∣∣ ≤ ∫ |f |.Proposition 5.0.9. Let the nonnegative function f be integrable over E. Then f isfinite a.e. on E.

Proposition 5.0.10. Let f be a nonnegative measurable function on E. Then∫E

f = 0 if and only if f = 0 a.e. on E.

Proposition 5.0.11. Let f be integrable over E. Then f is finite a.e. on E and∫E

f =

∫E−E0

f if E0 ⊂ E and m(E0) = 0

10

Chapter 6

Limit Theorems

Recall the following from calculus:

Definition 6.0.1. Let f be a real-valued function defined on a set E of real num-bers. We say that f is continuous at x in E provided that for each ε > 0, there isa δ(ε, x) > 0 for which

if x′ ∈ E and |x′ − x| < δ, then |f(x′)− f(x)| < ε.

The function f is said to be continuous on E provided it is continuous at eachpoint in its domain E.

Definition 6.0.2. A real-valued function f defined on a set E of real numbers issaid to be uniformly continuous provided for each ε > 0, there is a δ(ε) > 0such that for all x, x′ ∈ E,

if |x− x′| < δ, then |f(x)− f(x′)| < ε.

Definition 6.0.3. For a sequence {fn} of functions with common domain E, afunction f on E and a subset A of E, we say that

(i) The sequence {fn} converges pointwise on A provided for each x ∈ Aand ε > 0, there is an index N(x, ε) for which

|fn − f | < ε on A for all n ≥ N.

(ii) The sequence {fn} converges to f pointwise a.e. on A provided it con-verges to f pointwise on A−B, where m(B) = 0.

(iii) The sequence {fn} converges to f uniformly on A provided for eachε > 0, there is an index N(ε) for which

|fn − f | < ε on A for all n ≥ N.

(iv) For the sequence {fn} of measurable functions on E and f measurablefor which f and fn are finite a.e. on E, the sequence {fn} is said to converge inmeasure on E to f provided for each ε > 0,

limn→∞

m{x ∈ E : |fn(x)− f(x)| > ε} = 0

11

CHAPTER 6. LIMIT THEOREMS

Definition 6.0.4. A sequence of real numbers {an} is said to be Cauchy pro-vided for each ε > 0, there is an index N for which

if n,m ≥ N , then |am − an| < ε.

Theorem 6.0.5. (The Extreme Value Theorem) A continuous real-valued func-tion on a nonempty closed, bounded set of real numbers takes a minimum andmaximum value.

Theorem 6.0.6. (The Intermediate Value Theorem) Let f be a continuous real-valued function on the closed, bounded interval [a, b] for which f(a) < c < f(b).Then there is a point x0 ∈ (a, b) at which f(x0) = c.

Theorem 6.0.7. (The Bolzano-Weierstrass Theorem) Every bounded sequenceof real numbers has a convergent subsequence.

Theorem 6.0.8. (The Monotone Convergence Criterion) A monotone sequenceof real numbers converges if and only it is bounded.

Theorem 6.0.9. (The Cauchy Convergence Criterion) A sequence of real num-bers converges if and only if it is Cauchy.

Proposition 6.0.10. (i) Suppose µ is a finite measure. If fn → f a.e., thenfn converges to f in measure. (ii) If µ is a measure, not necessarily finite, andfn → f in measure, there is a subsequence such that fnj

→ f a.e.

Lemma 6.0.11. (Chebyshev’s Inequality) If 1 ≤ p <∞, then

µ({x : |f(x)| ≥ a}) ≤ 1

ap

∫|f |p dµ.

Lemma 6.0.12. (Fatou’s Lemma) Let {fn} be a sequence of nonnegative mea-surable functions on E.

If {fn} → f pointwise a.e. on E, then∫E

lim inf fn ≤ lim inf

∫E

fn.

Proposition 6.0.13. Let {fn} be a sequence of bounded measurable functionson a set of finite measure E.

If{fn} → f uniformly on E, then limn→∞

∫E

fn =

∫E

f.

Theorem 6.0.14. (Monotone Convergence Theorem) Let {fn} be an increasingsequence of nonnegative measurable functions on E.

If {fn} → f pointwise a.e. on E, then limn→∞

∫E

fn =

∫E

f.

Theorem 6.0.15. (Dominated Convergence Theorem) Let {fn} be a sequence ofmeasurable functions on E. Suppose there is a function g that is integrable overE and dominates {fn} on E in the sense that |fn| ≤ g on E for all n.

If {fn} → f pointwise a.e. on E, then f is integrable over E and limn→∞

∫E

fn =

∫E

f.

12

Chapter 7

Differentiation and Integration

Definition 7.0.1. A function f is differentiable at x if

limh→0

f(x+ h)− f(x)h

exists, and the limit is called the derivative of f at x and denoted f ′(x). Iff : [a, b]→ R, we say f is differentiable on [a, b] if the derivative exists for eachx ∈ (a, b) and both

limh→0+

f(a+ h)− f(a)h

and limh→0−

f(b+ h)− f(b)h

exist.

Theorem 7.0.2. Let f be a monotone function on the open interval (a, b). Then fis continuous except at a countable number of points in (a, b).

Theorem 7.0.3. (Lebesgue’s Theorem) If the function f is monotone on the openinterval (a, b), then it is differentiable almost everywhere on (a, b).

Definition 7.0.4. Let f be a real valued function defined on the closed, boundedinterval [a, b] and P = {x0, . . . , xk} be a partition [a, b]. Define the variationof f with respect to P by

V (f, P ) =k∑i=1

| f(xi)− f(xi−1) | .

Definition 7.0.5. Let f be a real-valued function defined on the closed, boundedinterval [a, b] and P = {x0, . . . , xk} be a partition of [a, b]. Define the totalvariation of f with respect to P by

TV (f) = sup{V (f, P ) : P a partition of [a, b]}.Definition 7.0.6. A real-valued function f on the closed, bounded interval [a, b]is said to be of bounded variation on [a, b] provided

TV (f) <∞

13

CHAPTER 7. DIFFERENTIATION AND INTEGRATION

Corollary 7.0.7. If the function f is of bounded variation on the closed, boundedinterval [a, b], then it is differentiable almost everywhere on the open interval(a, b) and f ′ is integrable over [a, b].

Definition 7.0.8. A real-valued function f on a closed, bounded interval [a, b] issaid to be absolutely continuous on [a, b] provided for each ε > 0, there is aδ > 0 such that for every finite disjoint collection {(ak, bk)}nk=1 of open intervalsin (a, b),

ifn∑k=1

[bk − ak] < δ thenn∑k=1

|f(bk)− f(ak)| < ε.

Definition 7.0.9. A real-valued function f is Lipschitz with constant M if

|f(x)− f(y)| ≤M |x− y|

for all x, y ∈ R.

A function f is Lipschitz with constant M if and only if f is absolutely contin-uous and |f ′| ≤M a.e.

Proposition 7.0.10. If the function f is Lipschitz on a closed, bounded interval[a, b], then it is absolutely continuous on [a, b].

Theorem 7.0.11. Let the function f be absolutely continuous on the closed,bounded interval [a, b]. Then f is the difference of increasing absolutely con-tinuous functions and, in particular, is of bounded variation.

Theorem 7.0.12. Let the function f be absolutely continuous on the closed,bounded interval [a, b]. Then f is differentiable almost everywhere on (a, b), itsderivative f ′ is integrable over [a, b] and∫ b

a

f ′ = f(b)− f(a).

Definition 7.0.13. We call a function f on a closed, bounded interval [a, b] theindefinite integral/antiderivative of g over [a, b] provided g is Lebesgue inte-grable over [a, b] and

f(x) = f(a) +

∫ x

a

g for all x ∈ [a, b].

Theorem 7.0.14. A function f on a closed, bounded interval [a, b] is absolutelycontinuous on [a, b] if and only if it is an indefinite integral over [a, b].

Theorem 7.0.15. Let f be integrable over the closed, bounded inteval [a, b]. Then

d

dx

∫ x

a

f = f(x) for almost all x ∈ (a, b).

14

CHAPTER 7. DIFFERENTIATION AND INTEGRATION

Definition 7.0.16. We say f is locally integrable if∫K|f(x)| dx is finite when-

ever K is compact.

Theorem 7.0.17. Let

fr(x) =1

m(B(x, r))

∫B(x,r)

f(y) dy.

If f is locally integrable, then fr(x)→ f(x) a.e. as r → 0.

Theorem 7.0.18. If F : R→ R is increasing, then F ′ exists a.e. and∫ b

a

F ′(x) dx ≤ F (b)− F (a)

whenever a < b.

15

CHAPTER 7. DIFFERENTIATION AND INTEGRATION

16

Chapter 8

Product Measures

Throughout this section (X,A, µ) and (Y,B, v) are two reference measure spaces.Consider the Cartesian product X × Y of X and Y. If A ⊂ X and B ⊂ Y , wecall A × B a rectangle. If A ∈ A and B ∈ B, we call A × B a measurablerectangle.

Definition 8.0.1. LetR be the collection of measurable rectangles contained inX × Y , then µ× ν is the product measure defined onR by

µ× ν(A×B) = µ(A) · ν(B) for A×B ∈ R.

Definition 8.0.2. Let E be a subset of X × Y and f a function on E. For a pointx ∈ X we define the x-section of E to be the set

Ex = {y ∈ Y : (x, y) ∈ E} ⊂ Y

and the function f(x, ·) defined onEx by f(x, ·)(y) = f(x, y) to be the x-sectionof f. The y-section is defined similarly.

Theorem 8.0.3. (Fubini-Tonelli Theorem) Suppose f : X × Y → R is measur-able with respect toA×B If either (a) f is nonnegative, or (ii)

∫|f(x, y)| d(µ×

ν)(x, y) <∞, then(i) for each x, the function y 7→ f(x, y) is measurable with respect to B;(ii) for each y, the function x 7→ f(x, y) is measurable with respect to A;(iii) the function g(x) =

∫f(x, y) ν(dy) is measurable with respect to A;

(iv) the function h(y) =∫f(x, y)µ(dx) is measurable with respect to B;

(v) we have∫f(x, y) d(µ×ν)(x, y) =

∫ [∫f(x, y) dµ(x)

]dν(y) =

∫ [∫f(x, y) dν(y)

]µ(dx).

17

CHAPTER 8. PRODUCT MEASURES

18

Chapter 9

Signed Measures

Definition 9.0.1. By a signed measure ν on the measurable space (X,M) wemean an extended real-valued set function ν : M → [−∞,∞] that possessesthe following properties:

(i) ν assumes at most one of the values +∞,−∞.(ii) ν(∅) = 0.(iii) For any countable collection {Ek}∞k=1 of disjoint measurable sets,

ν(∞⋃k=1

Ek) =∞∑k=1

ν(Ek).

Definition 9.0.2. Let ν be a signed measure. We say that a set A is positive (resp.negative) with respect to ν, provided A is measurable and for every subset E ofA we have ν(E) ≥ 0 (resp. ν(E) ≤ 0). A measurable set is called null withrespect to ν provided every measurable subset of it has ν measure zero.

Theorem 9.0.3. (The Hahn Decomposition Theorem) Let ν be a signed measureon the measurable space (X,M). Then there is a positive set A for ν and anegative set B for ν for which

X = A ∪B and A ∩B = ∅.

Definition 9.0.4. A decomposition of X into the union of two disjoint sets A andB for which A is positive for ν and B is negative is called a Hahn decompositionfor ν. Such a decomposition may not be unique.

Definition 9.0.5. If {A,B} is a Hahn decomposition for ν, then we define twomeasures ν+ (the positive part) and ν− (the negative part) with ν = ν+− ν− bysetting

ν+(E) = ν(E ∩ A) and ν−(E) = −ν(E ∩B).

Definition 9.0.6. Two measures ν1 and ν2 on (X,M) are said to be mutuallysingular (denoted ν1 ⊥ ν2) if there are disjoint measurable sets A and B withX = A ∪ B for which ν1(A) = ν2(B) = 0. The measures ν+ and ν− definedabove are mutually singular.

19

CHAPTER 9. SIGNED MEASURES

Theorem 9.0.7. (The Jordan Decomposition Theorem) Let ν be a signed mea-sure on the measurable space (X,M). Then there are two mutually singularmeasures ν+ and ν− on (X,M) for which ν = ν+−ν−. Moreover, there is onlyone such pair of mutually singular measures.

Definition 9.0.8. The decomposition of a signed measure ν given by this theoremis called a Jordan decomposition of ν.

Definition 9.0.9. The measure |ν| = ν+ + ν− is called the total variation mea-sure of ν.

20

Chapter 10

Lp Spaces

Definition 10.0.1. Let X be a linear space. A real-valued functional ‖ · ‖ on X iscalled a norm provided for each f and g in X and each real number a,

‖f + g‖ ≤ ‖f‖+ ‖g‖‖af‖ = |a|‖f‖

‖f‖ ≥ 0 and ‖f‖ = 0 if and only if f = 0

Definition 10.0.2. A normed linear space is a linear space together with anorm. If X is a linear space normed by ‖ · ‖ we say that a function in X is a unitfunction provided ‖f‖ = 1. For any f ∈ X, f 6= 0, the function f/‖f‖ is a unitfunction: it is a scalar multiple of f which we call the normalization of f.

Definition 10.0.3. For E measurable and for 1 ≤ p <∞, we define Lp(E) to bethe collection of all functions f for which∫

E

|f |p <∞

Definition 10.0.4. Let (X,A, µ) be a σ-finite measure space. For 1 ≤ p < ∞,define the Lp norm of f by

‖f‖p =(∫|f(x)|p dµ

) 1p

Definition 10.0.5. We call a function f essentially bounded on E, provided thereis some M ≥ 0, called an essential upper bound for f, for which

|f(x)| ≤M for almost all x ∈ E.

Definition 10.0.6. For E measurable, we define L∞(E) to be the collection ofall functions which are essentially bounded.

Definition 10.0.7. The L∞ norm of a function f is the smallest M such that|f | ≤M a.e. Or,

‖f‖∞ = inf{M : µ({x : |f(x)| ≥M}) = 0}

21

CHAPTER 10. LP SPACES

Definition 10.0.8. For 1 ≤ p < ∞, define the normed linear space `p to be thecollection of real sequences a = (a1, a2, . . .) for which

∞∑k=1

|ak|p <∞ with norm being ‖a‖p =

(∞∑k=1

|ak|p) 1

p

Similarly, we define the normed linear space `∞ to be the collection of all realbounded sequences. For a sequence {ak} in `∞, define the norm to be

‖{ak}‖∞ = sup1≤k<∞|ak|.

Definition 10.0.9. The conjugate of a number p ∈ (1,∞) is the number q =p/(p− 1), which is the unique number q ∈ (1,∞) for which

1

p+

1

q= 1

The conjugate of 1 is defined to be∞ and the conjugate of∞ is defined to be 1.

Lemma 10.0.10. (Young’s Inequality) For 1 < p <∞, q the conjugate of p, andany two positive numbers a and b,

ab ≤ ap

p+bq

q.

Lemma 10.0.11. (Holder’s Inequality) Let 1 < p, q < ∞ such that 1/p +1/q = 1. Then for any Lebesgue measurable functions f and g on a Lebesguemeasurable set E, we have ∫

E

|f · g| ≤ ‖f‖p · ‖g‖q

Lemma 10.0.12. (Minkowski’s Inequality) Let 1 ≤ p < ∞. Then for anyLebesgue measurable functions f and g on a Lebesgue measurable set E, wehave

‖f + g‖p ≤ ‖f‖p + ‖g‖p.

Corollary 10.0.13. Let E be a measurable set of finite measure and 1 ≤ p1 <p2 ≤ ∞. Then Lp2(E) ⊂ Lp1(E).

Definition 10.0.14. A sequence {fn} in a linear space X that is normed by ‖ · ‖is said to converge to f in X provided

limn→∞

‖fn − f‖ = 0.

We write{fn} → f or lim

n→∞fn = f in X

to mean that each fn and f belong to X and limn→∞ ‖fn − f‖ = 0.

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CHAPTER 10. LP SPACES

For a sequence {fn} and function f in Lp(E), {fn} → f in Lp(E) if and onlyif

limn→∞

∫E

|fn − f |p = 0.

Proposition 10.0.15. If fn converges to f in Lp, then it converges in measure.

Definition 10.0.16. A sequence {fn} in a linear space X that is normed by ‖ · ‖is said to be Cauchy in X provided for each ε > 0, there is a natural number Nsuch that

‖fn − fm‖ < ε for all m,n ≥ N.

A normed linear space X is said to be complete provided every Cauchy sequencein X converges to a function in X. A complete normed linear space is called aBanach space.

Proposition 10.0.17. Let X be a normed linear space. Then every convergentsequence in X is Cauchy. Moreover, a Cauchy sequence in X converges if it hasa convergent subsequence.

Theorem 10.0.18. (The Riesz-Fischer Theorem) Let E be a measurable set and1 ≤ p ≤ ∞. Then Lp(E) is a Banach space. Moreover, if {fn} → f in Lp(E),a subsequence of {fn} converges pointwise a.e. on E to f.

Theorem 10.0.19. Let E be a measurable set and 1 ≤ p <∞. Suppose {fn} isa sequence in Lp(E) that converges pointwise a.e. on E to the function f whichbelongs to Lp(E). Then

{fn} → f in Lp(E) if and only if limn→∞

∫E

|fn|p =∫E

|f |p.

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CHAPTER 10. LP SPACES

24

Chapter 11

The Radon-Nikodym Theorem

Definition 11.0.1. A measure ν is said to be absolutely continuous with respectto a measure µ if ν(A) = 0 whenever µ(A) = 0. We write ν � µ.

Theorem 11.0.2. (The Radon-Nikodym Theorem) Let (X,M, µ) be a σ-finitemeasure space and ν a σ-finite measure defined on the measurable space (X,M)that is absolutely continuous with respect to µ. Then there is a nonnegative func-tion f on X that is measurable with respect toM for which

ν(E) =

∫E

f dµ for all E ∈M.

The function f is unique in the sense that if g is any nonnegative measurablefunction on X that also has this property, then g = f a.e.

Theorem 11.0.3. (The Lebesgue Decomposition Theorem) Let (X,M, µ) be aσ-finite space and ν a σ-finite measure on the measurable space (X,M). Thenthere is a measure ν0 onM, singular with respect to µ, and a measure ν1 onM,absolutely continuous with respect to µ, for which ν = ν0+ ν1. The measures ν0and ν1 are unique.

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CHAPTER 11. THE RADON-NIKODYM THEOREM

26

Chapter 12

Linear Operators

Definition 12.0.1. A linear functional on a linear space X is a real-valued func-tion T on X such that for g and h in X and a and b real numbers,

T (a · g + b · h) = a · T (g) + b · T (h).

Definition 12.0.2. For a normed linear space X, a linear functional T on X issaid to be bounded provided there is an M ≥ 0 for which

|T (f)| ≤M · ‖f‖ for all f ∈ X.

The infimum of all such M is called the norm of T and denoted ‖T‖∗. Also,‖T‖∗ = sup{T (f) : f ∈ X, ‖f‖ ≤ 1}.

Definition 12.0.3. Let X be a normed linear space. Then the collection ofbounded linear functionals on X is a linear space on which ‖ · ‖∗ is a norm.This normed linear space is called the dual space of X and denoted X∗.

Proposition 12.0.4. Let E be a measurable set, 1 ≤ p < ∞, q the conjugate ofp, and g belong to Lq(E). Define the functional T on Lp(E) by

T (f) =

∫E

g · f for all f ∈ Lp(E).

Then T is a bounded linear functional on Lp(E) and ‖T‖∗ = ‖g‖q.

Theorem 12.0.5. (The Riesz Representation Theorem for the Dual of Lp(E))Let E be a measurable set, 1 ≤ p < ∞, and q the conjugate of p. For eachg ∈ Lq(E), define the bounded linear functionalRg on Lp(E) by

Rg(f) =

∫E

g · f for all f ∈ Lp(E).

Then for each bounded linear functional T on Lp(E), there is a unique functiong ∈ Lq(E) for which

Rg = T, and ‖T‖∗ = ‖g‖q.

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CHAPTER 12. LINEAR OPERATORS

Definition 12.0.6. Let X and Y be linear spaces. A linear operator is a mappingT : X → Y provided for each u, v ∈ X , and real numbers a and b,

T (au+ bv) = aT (u) + bT (v).

Definition 12.0.7. Let X and Y be normed linear spaces. A linear operatorT : X → Y is said to be bounded provided there is a constantM ≥ 0 for which

‖T (u)‖ ≤M‖u‖ for all u ∈ X.

Then infimum of all such M is called the operator norm of T and denoted ‖T‖.The collection of bounded linear operators from X to Y is denoted by L(X, Y ).

Theorem 12.0.8. A linear operator between normed linear spaces is continuousif and only if it is bounded.

Proposition 12.0.9. Let X and Y be normed linear spaces. Then the collectionof bounded linear operators from X to Y, L(X, Y ), is a normed linear space.

Theorem 12.0.10. Let X and Y be normed linear spaces. If Y is a Banach space,then so is L(X, Y ).

Definition 12.0.11. A mapping f : X → Y from the topological space X to thetopological space Y is said to be open provided the image of each open set in Xis open in the topological space f(X).

Theorem 12.0.12. (The Open Mapping Theorem) Let X and Y be Banach spacesand the linear operator T : X → Y be continuous. Then its image T (X) is aclosed subspace of Y if and only if the operator T is open.

Definition 12.0.13. The graph of a mapping T : X → Y is the set {(x, T (x)) ∈X × Y : x ∈ X}. Therefore an operator is closed if and only if its graph is aclosed subspace of the product space X × Y .

Theorem 12.0.14. (The Closed Graph Theorem) Let T : X → Y be a linearoperator between Banach spaces X and Y. Then T is continuous if and only if itis closed.

Theorem 12.0.15. (The Uniform Boundedness Principle) For X a Banach spaceand Y a normed linear space, consider a family F ⊂ L(X, Y ). Suppose thefamilyF is pointwise bounded in the sense that for each x in X there is a constantMx ≥ 0 for which

‖T (x)‖ ≤Mx for all T ∈ F .Then the family F is uniformly bounded in the sense that there is a constantM ≥ 0 for which ‖T‖ ≤M for all T in F .

Theorem 12.0.16. (The Banach-Steinhaus Theorem) Let X be a Banach space,Y a normed linear space, and {Tn : X → Y } a sequence of continuous linearoperators. Suppose that for each x ∈ X ,

limn→∞

Tn(x) exists in Y.

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CHAPTER 12. LINEAR OPERATORS

Then the sequence of operators {Tn : X → Y } is uniformly bounded. Further-more, the operator T : X → Y defined by

T (x) = limn→∞

Tn(x) for all x ∈ X

is linear, continuous, and

‖T‖ ≤ lim inf ‖Tn‖.

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CHAPTER 12. LINEAR OPERATORS

30

Chapter 13

Hilbert Spaces

Definition 13.0.1. Let H be a linear space. A function 〈·, ·〉 : H × H → R iscalled an inner product on H provided for all x1, x2, x and y ∈ H and realnumbers a and b,

(i) 〈ax1 + bx2, y〉 = a〈x1, y〉+ b〈x2, y〉,(ii) 〈x, y〉 = 〈y, x〉,(iii) 〈x, x〉 > 0 if x 6= 0.A linear space H together with an inner product on H is called an inner

product space.

Lemma 13.0.2. (The Cauchy-Schwarz Inequality) For any two vectors u, v in aninner product space H,

|〈u, v〉| ≤ ‖u‖ · ‖v‖.

Proposition 13.0.3. For a vector h in an inner product space H, define

‖h‖ =√〈h, h〉.

Then ‖ · ‖ is a norm on H called the norm induced by the inner product 〈·, ·〉.

Lemma 13.0.4. (The Parallelogram Identity) For any two vectors u, v in aninner product space H,

‖u− v‖2 + ‖u+ v‖2 = 2‖u‖2 + 2‖v‖2.

Definition 13.0.5. An inner product space H is called a Hilbert space providedit is a Banach space with respect to the norm induced by the inner product.

Definition 13.0.6. Two vectors u, v in the inner product space H are said tobe orthogonal provided 〈u, v〉 = 0. A vector u is said to be orthogonal to asubset S of H provided it is orthogonal to each vector in S. We denote by S⊥ thecollection of vectors in H that are orthogonal to S.

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CHAPTER 13. HILBERT SPACES

32