Some Historical developments of real analysis Weierstrass's nowhere 1872 differentiable function Introduction of 1881 functions by Jordan and 1883 Cantor set later connection with rectifiability 1890 Space filling curve by Peano 1898 Borel's measurable sets 1902 bebesgue's theory of Consfruction 1905 measure and integration non-measurable sefs by Vitali 2
2 Some Historical developments of real analysis Weierstrass’s nowhere differentiable function 1872 Introduction of BV functions by Jordan and later connection with rectifiability Cantor set Space filling curve by Peano Construction of non-measurable sets by Vitali Borel’s measurable sets Lebesgue’s theory of measure and integration 1881 1883 1890 1898 1902 1905
8 CHAPTER 1.PRELIMINARY The othtThererexpectd oction of Fourer 1.2 Cardinality In following sections,we establishsome foundations on the set theory and the topology and geometry the E t this miliar with infinite elements?This requires the concept of the cardinality of a set For two sets with finite number of elements,it is clear which set contains more elements.For two sets with in me e if V2 B.th f()=2.A mapf:ABis called a bijection if f is both injective and surjective.Clearly,a map f:AB has a well-defined inverse,if and only if f is a bijection B are called to tion The cardinal number of natural numbers N is denoted by o.(Countable) Erample 1.Each infinite set contains a countable subset Erample 2.Countable uion of countable sets is countable. Proof.Array this union as an infinite square,and enumerate in a zigzag way. 0 Erample 3.All rational numbers Q is countable. Erample 4.Finite cartesian product of countable sets is countable Proof.Visualize this union as an infinitek dimensional cube,and enumerate in a zigzag way. ◇ Erample 5.The set of all real numbers R is not countable. (0.1l is not countable.We accept each real n om position.That we write0.25as0.249999999,1as0.99999. ete Now suppos (0,1]is countabl then we have an en meration for all mbers in(0,，小say that y is in The cardinality ofR is called The decimal representation shows that countable product of finite sets has cardinal number Erample 6.R,(0,1],(0,1],R all have same cardinal number Theorem 1.1.There docs not erist marimal cardinal number Proof.Given any set A.consider its power set 24.namely the set of all subsets of A.We ca show they have different cardinality.Otherwise,there exists a bijectionf:A24,where f(a) corresponds to a subset of A.Define a subset of A as follows: B={xz度f(z
8 CHAPTER 1. PRELIMINARY The second part begins with the rudiment of the function spaces, followed by an introduction to Fourier analysis. We study both Fourier series and Fourier transform together with their applications. The connection with real analysis is intimacy. There are also many unexpected connections of Fourier analysis to wide-ranging mathematical topics such as Number theory, Discrete geometry, Probability theory. We convey to the reader only a small portion of this fascinating subject. 1.2 Cardinality In following sections, we establish some foundations on the set theory and the topology and geometry of the Euclidean space. We assume the reader is familiar with basic notions of sets, operations between sets, etc. In this section, we address the following question: how to compare two sets with infinite elements? This requires the concept of the cardinality of a set. For two sets with finite number of elements, it is clear which set contains more elements. For two sets with infinite elements, which contains ’more’ elements relies on the mappings between them. A map f : A → B is an assignment to each element of A a unique element in B. f is called injective, if f(x) 6= f(y), for x 6= y. f is called surjective if ∀z ∈ B, there exists x ∈ A such that f(x) = z. A map f : A → B is called a bijection if f is both injective and surjective. Clearly, a map f : A → B has a well-defined inverse, if and only if f is a bijection. A and B are called to have same cardinality if there exists a bijection f : A → B, denoted by A ∼ B. Sometimes, we shall refer to the cardinal number of a set A, denoted by A ¯¯. The cardinal number of natural numbers N is denoted by ℵ0. (Countable) Example 1. Each infinite set contains a countable subset. Example 2. Countable union of countable sets is countable. Proof. Array this union as an infinite square, and enumerate in a zigzag way. Example 3. All rational numbers Q is countable. Example 4. Finite cartesian product of countable sets is countable. Proof. Visualize this union as an infinite k dimensional cube, and enumerate in a zigzag way. Example 5. The set of all real numbers R is not countable. Proof. We prove (0, 1] is not countable. We accept each real number in (0, 1] has a decimal representation, which is unique if we don’t allow the appearance of all zeros after some position. That is we write 0.25 as 0.249999999..., 1 as 0.99999...., etc. Now suppose (0, 1] is countable, then we have an enumeration for all numbers in (0, 1], say 0.a11a12a13...., 0.a21a22a23..., ... We can choose bii ∈ {0, 1, 2, ..., 9} \ aii, for each i. Let y = 0.b11b22b33..., a moment of thought shows that y is indeed not in the enumeration list. A contradiction. The cardinality of R is called ℵ1. The decimal representation shows that countable product of finite sets has cardinal number ℵ1. Example 6. R, (0, 1], [0, 1], R n all have same cardinal number ℵ1. Theorem 1.1. There does not exist maximal cardinal number. Proof. Given any set A, consider its power set 2A, namely the set of all subsets of A. We can show they have different cardinality. Otherwise, there exists a bijection f : A → 2 A, where f(a) corresponds to a subset of A. Define a subset of A as follows: B = {x|x /∈ f(x)}
1.3.TOPOLOGY OF THE EUCLIDEAN SPACE Now an is B=f(r)for mex∈A? This of the barber paradox which was raised by Bertrand Russellas There is no set whose cardinality is strictly between that of the integers and the real numbers. ipedia. 1.3 Topology of the Euclidean space We use Ra for n-dimensional Euclidean space.For =(1..andy=(.),the inner product is defined as 工·y=r1边+r22+··+xmn Norm is defined as 回=√+…+ Open ball centered at r of radius r is denoted by B(.r).i.e., B(z.r)=uy-<r). Closed ball is B(r)=fly-). ope a))cod eabe s open ha ce is of the form is called an set if ove int of A is an interio if ()for.The uion of ts acculation points is called the denoted by A set A is called closed,if A is an n se existsR> nice property of being a compact setis that any open cover hasa finite subcover. Theneml3但eineBare.AcRrwaompdtfamdomtfeenyopeacorgofAcontains We also recall the theorem of nested closed sets. Theorem 1.4.Let AAA..be a sequence of nested non-emply closed sets.Then ∩e1A≠0. nle 8 (Cantor set).Let co =o 11 the unit closed int moval of the middle ne third open intervals of each connected components of Cn-1.For example,C2=0,1]. c=∩c. is the Cantor set
1.3. TOPOLOGY OF THE EUCLIDEAN SPACE 9 Now an amusing question confronts us: is B = f(x) for some x ∈ A? This proof is reminiscent of the barber paradox, which was raised by Bertrand Russell as follows: a barber in a town claims to be the ”one who shaves all those, and those only, who do not shave themselves.” The question is, does the barber shave himself? Remark 1.2 (Continuum hypothesis). Cantor in 1878 raised the following hypothesis concerning the size of infinite sets: There is no set whose cardinality is strictly between that of the integers and the real numbers. Establishing its truth or falsehood is the first of Hilbert’s 23 problems presented in 1900. The reader is referred to https://en.wikipedia.org/wiki/Continuum hypothesis for a thorough introduction. 1.3 Topology of the Euclidean space We use R n for n-dimensional Euclidean space. For x = (x1, · · · , xn and y = (y1, · · · , yn), the inner product is defined as x · y = x1y1 + x2y2 + · · · + xnyn. Norm is defined as |x| = q x 2 1 + · · · + x 2 n . Open ball centered at x of radius r is denoted by B(x, r), i.e., B(x, r) = {y||y − x| < r}. Closed ball is B(x, r) = {y||y − x| ≤ r}. An open cube is of the form (a1, b1)×(a2, b2)×· · ·×(an, bn), closed cube is [a1, b1]×· · ·×[an, bn]. A half-open half-closed cube is of the form (a1, b1] × · · · × (an, bn]. Given A ⊂ R n, x is called an interior point of A if there exists r > 0 such that B(x, r) ⊂ A. A is called an open set, if every point of A is an interior point. x is called an accumulation point of A, if (B(x, r) \ {x}) ∩ A 6= ∅, for all r > 0. The union of A with its accumulation points is called the closure of A, denoted by A¯. A set A is called closed, if A¯ is an open set. A family of open sets {Oα}α∈Λ is called an open cover of A if A ⊂ S α Oα. A is bounded if there exists R > 0, such that A ⊂ B(0, R). A set is called compact if it is both bounded and closed. A nice property of being a compact set is that any open cover has a finite subcover. Theorem 1.3 (Heine-Borel). A ⊂ R n is a compact set if and only if every open cover of A contains a finite subcover. We also recall the theorem of nested closed sets. Theorem 1.4. Let A1 ⊃ A2 ⊃ · · · ⊃ An ⊃ · · · be a sequence of nested non-empty closed sets. Then T∞ n=1 An 6= ∅. B ⊂ A is called dense in A, if B¯ = A. A is called nowhere dense if there exists no interior point of A¯. Example 7. Take r /∈ Q, < x > denotes the fractional part of x. Then {< rn >}n=1,2,··· is dense in [0, 1]. Example 8 (Cantor set). Let C0 = [0, 1] the unit closed interval. C1 = [0, 1 3 ] ∪ [ 2 3 , 1], the removal of the middle 1 3 open interval from C0. Cn is obtained inductively by removing the middle one third open intervals of each connected components of Cn−1. For example, C2 = [0, 1 9 ]∪[ 2 9 , 1 3 ]∪[ 2 3 , 7 9 ]∪[ 8 9 , 1]. C := \∞ n=0 Cn is the Cantor set
CHAPTER 1.PRELIMINARY 国 翻铜 Figure 1.1:Cantor set The following proposition lists several properties of the Cantor set. Proposition 1.5.The Cantor set C defined as above is non-empty and satisfies the following properttes. ·C is closed. .C does not contain any interior point,hence it is nowhere dense. .C is uncountable,and its cardinal number is closed. Suppose C is an interior point,then there exists>0,such that (+)CC.Taking N large enoug such that her with clo cmC.folo here dens Using the decimal representation of base 3 for all real numbers in [0,1],i.e,=,where inwedon'tall the tha di= the has the cardinal number 1.4 Metric space and Baire Category theorem Given a set X,a map d:xR satisfying 1.Symmetry d(,)=d(y,); 2.Positivity d()0 and =holds if and only if=y: 3.Triangle inequality d(z,)+d(,z)≥d(工，z is called a metric on X.(X,d)is then called a metric space. tion of convergence.limn=z if and only if limd(n,) Ve >0,there exists N,such that d(rn,rm)se,Vn,m N. A metric space is called complete if any Cauchy sequence is convergent in the space.The concepts of open balls,open sets,closed sets,interior points,closure,etc,all generalize to the metric space. Theorem 1.6(Baire Category Theorem).A non-emply complete metric space is not a counable DSimilaly D)s nonempty open set,can chocse suc that B
10 CHAPTER 1. PRELIMINARY Figure 1.1: Cantor set The following proposition lists several properties of the Cantor set. Proposition 1.5. The Cantor set C defined as above is non-empty and satisfies the following properties: • C is closed. • C does not contain any interior point, hence it is nowhere dense. • C is uncountable, and its cardinal number is ℵ1. Proof. C is not empty. A moment of thought shows that the end points of those middle third intervals all remain in C. Since each Cn is closed, the intersection of countable closed sets is still closed. Suppose x ∈ C is an interior point, then there exists δ > 0, such that (x − δ, x + δ) ⊂ C. Taking N large enough such that 1 3N < 2δ, it follows (x − δ, x + δ) is not contained in CN , as the length of each connected component of CN is 1 3N . This shows that C does not have any interior points. Together with closeness of C, it follows that C is nowhere dense. Using the decimal representation of base 3 for all real numbers in [0, 1], i.e, x = P∞ i=1 ai 3 i , where ai ∈ {0, 1, 2}. Again to ensure the uniqueness, we don’t allow the situation that ai = 0 ∀i ≥ N for some N, unless x = 0 which corresponds to ai = 0 for all i. The removal of the middle third intervals prevents the appearance of 1 in this decimal representation. Therefore C ∼ {0, 2} N which has the cardinal number ℵ1. 1.4 Metric space and Baire Category theorem Given a set X, a map d : X × X → R+ satisfying 1. Symmetry d(x, y) = d(y, x); 2. Positivity d(x, y) ≥ 0 and = holds if and only if x = y; 3. Triangle inequality d(x, y) + d(y, z) ≥ d(x, z); is called a metric on X. (X, d) is then called a metric space. Using metric, one can define the notion of convergence. limn→∞ xn = x if and only if limn→∞ d(xn, x) = 0. {xn} is called a Cauchy sequence, if ∀ > 0, there exists N, such that d(xn, xm) ≤ , ∀n, m > N. A metric space is called complete if any Cauchy sequence is convergent in the space. The concepts of open balls, open sets, closed sets, interior points, closure, etc, all generalize to the metric space. Theorem 1.6 (Baire Category Theorem). A non-empty complete metric space is not a countable union of nowhere dense sets. Proof. Suppose not. Then assume X = S∞ n=1 Dn, where each Dn is a nowhere dense set. Clearly X \ D1 is not empty, therefore there exists an interior point x1 and 1 > 0 such that B(x1, 1) ⊂ X \ D1. Similarly D2 c ∩ B(x1, ) is a nonempty open set, we can choose x2, 2 such that B(x2, 2) ⊂
1.5.CONTINUOUS FUNCTIONS AND DISTANCE IN METRIC SPACE 11 ce of nested balls B( )C B( 1).mor equences and it cor rerges to. Using the Baire category theorem,we get another proof that 0,I is uncountable. Gs set,countable union of closed sets is called an 0ion7.Phere dow not erist a mction加.广：R→hich t continous ony t7 We need a lemma first Lemma 1.8.The points of continuity of f is a Gs set. Proof.Recall that f is continuous at z if and only if the oscillation wf()=0.Therefore the set of points of continuity of f is ∩wr回)<》 It is easy to show that ()<is open. 0 Q=0G where each G is open set.We write QasQ=),then R=UG:Ufa). =1 neSCeckoad.saupoeteotainsanialaiaroil,thamtdaeethisanopemileralz）cG (z,°GmQ. The only ossible case is =u.Hence Ge is nowhere dens The above expression writes R as a union of countable nowhere dense sets.This contradicts to the Baire category theorem. 1.5 Continuous functions and Distance in metric space Given a function f:ECR"R,f is continuous at rE,if e0,there exists6>0 such that If(y)-fr川≤6，y∈B(x,6)nE. f is called continuous on E if f is continuous at every point of E.This definition does not require E is open. Theorem 1.9.Suppose f:FR be a continuous function defined on a compact set F,then f is uniform continuous and attains its marimum and minimum
1.5. CONTINUOUS FUNCTIONS AND DISTANCE IN METRIC SPACE 11 D2 c ∩ B(x1, ). Inductively, we get a sequence of nested balls B(xn, n) ⊂ B(xn−1, n−1), moreover we can easily arrange that limn→∞ n = 0. Thus {xn} is a Cauchy sequences and it converges to, say x. Since X = S∞ n=1 Dn, thus x ∈ Dk for some k. However due to the construction x ∈ B(xk, k), which contradicts to that B(xk, k) ∩ Dk = ∅. Using the Baire category theorem, we get another proof that [0, 1] is uncountable. Countable intersection of open sets is called a Gδ set, countable union of closed sets is called an Fσ set. We give a more interesting application of Baire’s category theorem. Proposition 1.7. There does not exist a function f : R → R which is continuous only at all rational numbers. We need a lemma first. Lemma 1.8. The points of continuity of f is a Gδ set. Proof. Recall that f is continuous at x if and only if the oscillation ωf (x) = 0. Therefore the set of points of continuity of f is \∞ n=1 {x|ωf (x) < 1 n }. It is easy to show that {x|ωf (x) < 1 n } is open. Proof of the Proposition. Using the above lemma, it is suffice to show that Q is not a Gδ set. Suppose not, then assume Q = \∞ n=1 Gn, where each Gn is open set. We write Q as Q = {q1, q2, · · · }, then R = [∞ n=1 G c n [∞ i=1 {qi}. Gc n is closed, suppose it contains an interior point, then there exists an open interval (x, y) ⊂ Gc n . Therefore (x, y) c ⊃ Gn ⊃ Q. The only possible case is x = y. Hence Gc n is nowhere dense. The above expression writes R as a union of countable nowhere dense sets. This contradicts to the Baire category theorem. 1.5 Continuous functions and Distance in metric space Given a function f : E ⊂ R n → R, f is continuous at x ∈ E, if ∀ > 0, there exists δ > 0 such that |f(y) − f(x)| ≤ , ∀y ∈ B(x, δ) ∩ E. f is called continuous on E if f is continuous at every point of E. This definition does not require E is open. Theorem 1.9. Suppose f : F → R be a continuous function defined on a compact set F, then f is uniform continuous and attains its maximum and minimum