MTH 303: Real Analysis

Semester 1, 2013-14

1. Real numbers 1.1. Decimal and ternary expansion of real numbers.

(i) [0, 1] and R are uncountable. (ii) Cantor set is an uncounbtable set of measure zero.

1.2. Order properties of real numbers. (i) Least upper bound axiom. (ii) Every subset of R that has a lower bound also has a greatest lower bound. (iii) Archimedean property. (iv) Density of Q.

2. Sequences of real numbers 2.1. Limits of sequences. (i) Sequences and their limits with examples. (ii) A sequence in R has at most one limit. (iii) A sequence converges if and only if its tail converges. (iv) Let (xn) be a sequence of real numbers and let x ∈ R. If (an) is a sequence of positive reals with lim(an) = 0 and if we have a constant C > 0 and some m ∈ N such that |xn − x| ≤ Can, for n ≥ m, then lim(xn) = x.

2.2. Limit Therorems. (i) Bounded sequences with examples. (ii) A convergent sequence of real numbers is bounded. (iii) Convergence of the sum, difference, product, and quotient of sequences. (iv) The limit of a non-negative sequence is non-negative.

(v) Squeeze Theorem: Suppose that X = (xn), Y = (yn), and Z = (zn) are sequences such that xn ≤ yn ≤ zn, ∀ n, and lim(xn) = lim(zn). Then Y converges and lim(xn) = lim(yn) = lim(zn).   (vi) Let X = (x ) be a sequence of positive reals such that L = lim xn+1 . n xn If L < 1, then X converges and lim(xn) = 0. 1 2

2.3. Monotone Sequences. (i) Monotone sequences with examples. (ii) Monotone Covergence Theorem: A sequence of real numbers is conver- gent if and only if it is bounded. Moreover,

(a) If X = (xn) is a bounded increasing sequence, then lim(xn) = sup{xn | n ∈ N}. (b) If Y = (yn) is a bounded decreasing sequence, then lim(yn) = inf{yn | n ∈ N}.

2.4. Subsequences. (i) Subsequences with examples. 0 (ii) If a sequence X = (xn) converges to x, then any subsequence X = (xnk) of X also converges to x.

(iii) Divergence Criteria: If a sequence X = (xn) of real numbershas either of the two properties, then X is divergent. (a) X has 2 convergent subsequences converging to different limits. (b) X is unbounded.

(iv) Monotone Subsequence Theorem: If X = (xn) is a sequence of real numbers , them there exists a subsequence of X that is monotone. (v) Bolzano-Weierstrass Theorem: Every bounded sequence of real numbers has a convergent subsequence.

(vi) If X = (xn) is a bounded sequence of reals such that every subsequence of X converges to x ∈ R, the X converges to x.

2.5. The Cauchy Criterion. (i) Cauchy sequences with examples. (ii) Every convergent sequence is Cauchy. (iii) Every Cauchy sequence is bounded. (iv) Completeness property: Every Cauchy sequence is convergent. (v) Contractive sequences. (vi) Every contractive sequence is Cauchy.

2.6. Properly divergent sequences. (i) Properly divergent sequences with examples. (ii) A monotone sequence of real numbers is properly divergent if and only if it is unbounded

(iii) Let (xn) and (yn) be sequences such that xn ≤ yn, for all n. (a) If lim(xn) = +∞, then lim(yn) = +∞. 3

(b) If lim(yn) = −∞, then lim(xn) = +∞. (iv) Let (xn) and (yn) be sequences of positive real numbers such that lim(xn/yn) = L, for some L > 0. Then lim(xn) = +∞ if and only if lim(yn) = +∞.

2.7. Limit inferior and superior. (i) Subsequence limit (ii) Limit inferior and superior

(iii) A bounded sequence (xn) is convergent if and only if lim sup(xn) = lim inf(xn).

3. Sequences of functions 3.1. Pointwise and uniform convergence. (i) Pointwise convergent sequences with examples. (ii) Uniformly convergent sequence with examples. (iii) Criterion for non-uniform convergence.

(iv) The uniform norm k kA. (v) A sequence (fn) of bounded functions on A ⊆ R converges uniformly on A to f if and only if kfn − fkA → 0. (vi) Cauchy criterion for uniform convergence: A sequence of bounded functions on A ⊆ R converges uniformly on A to a bounded function f if and only for each  > 0 there exists a number K ∈ N such that for all m, n ≥ K, then kfm − fnkA ≤ . (vii) Uniform Limit Theorem: Let (fn) be a sequence of continuous function on a set A ⊆ R such that fn converges to f uniformly on A to a function f. Then f is continuous on A.

(viii) Interchange of limit and derivative: Let (fn) be a sequence of functions on a bounded interval J ⊂ R. Suppose that fn converges pointwise at 0 x0 ∈ J and that (fn) exists on J and converges uniformly to a function 0 g. Then (fn) converges uniformly on J to a function f such that f = g. (ix) Compactness in R. (x) Dini’s Theorem: if (fn) is a sequence of continuous functions on [a, b] that converges on [a, b] to a continuous function f. Then the conver- gence of the sequence is uniform.

3.2. The metric space C[a, b]. (i) Brief introduction to metric spaces and norms. (ii) The supremum norm and the metric it induces in C[a, b]. 4

(iii) Convergence with respect to the supremum norm is equivalent to uniform convergence. (iv)C[ a, b] is complete. (v) Boundedness and total boundedness in metric spaces. (vi) A totally bounded subset of a metric space is bounded. (vii) -dense subsets of a metric space. (viii) A subset A of metric space (X, d) is totally bounded if and only if for every  > 0, there exists a finite subset of A that is -dense in A. (ix) A subset A of a metric space (X, d) is totally bounded if and only if every sequence in A has a Cauchy subsequence. (x) Equicontinuous families of functions. (xi) Ascoli-Arzela Theorem: Let F be a bounded equicontinuous subset of the metric space C[a, b]. Then F is totally bounded. (xii) Every bounded equicontinuous sequence of functions in C[a, b] has a uniformly convergent subsequence. (xiii) Weierstrass Approximation Theorem: Let f be any function in C[a, b]. Then given  > 0, there exists a polynomial P such that kP − fk < . In other words, the set P of polynomials is dense in C[a, b].

4. Series of functions 4.1. Convergence and uniform convergence of a series of functions. (i) Convergence and uniform convergence with examples.

(ii) Uniform Limit Theorem: Let (fn) be a sequence of real-valed function P∞ on a metric space X. If n=1 fn converges uniformly to f on X and if each fn is continous at a ∈ X, then f is also continuous at X. P∞ (iii) Weierstrass M-test: Let n=1 fn be a series of real-valued functions on a set E. If there exists a sequence (Mn) of positive real numbers P∞ P∞ P∞ with k=1 Mk < ∞ such that k=1 fk(x)  k=1 Mk, for all x ∈ E, P∞ then k=1 fk converges uniformly on E. P∞ k (iv) If the power series k=0 akx coverges for x = x0, then it converges uniformly on [−a, a], where a is any number such that 0 < a < |x0|. P∞ (v) Dini’s Theorem: Let n=1 fn be a series of continuous nonnegative P∞ functions on the a compact metric space X. If n=1 fn converges on X to the continuous function f, then the convergence is uniform on X.

4.2. Taylor’s Theorem. (i) Taylor series with examples. 5

(ii) Taylor’s Theorem (with intergal form of remainder): Let f be a real valued function on [a, a + h] such that f (n+1)(x) exists for every x ∈ [a, a + h] and f (n+1) is continuous in [a, a + h]. Then f 0(a) f 00(a) f(x) = f(a) + (x − a) + (x − a)2 + ... 1! 2! f (n)(a) + (x − a)n + R (x ∈ [a, a + h]), n! n+1 where Z x 1 n (n+1) Rn+1 = (x − t) f (t) dt . n! a (iii) Taylor’s Theorem (with Lagrange form of remainder): Let f be a real valued function on [a, a + h] such that f (n+1)(x) exists for every x ∈ [a, a+h] and f (n+1) is continuous in [a, a+h]. Then if x ∈ [a, a+h], there exists a number c ∈ [a, x] such that f 0(a) f 00(a) f (n)(a) f(x) = f(a) + (x − a) + (x − a)2 + ... + (x − a)n 1! 2! n! f (n+1)(c) + (x − a)n+1. (n + 1)! (iv) Taylor’s Theorem (with Cauchy form of remainder): Let f be a real valued function on [a, a + h] such that f (n+1)(x) exists for every x ∈ [a, a + h] and f (n+1) is continuous in [a, a + h]. Then if x ∈ [a, a + h], there exists a number c ∈ [a, x] such that f 0(a) f 00(a) f (n)(a) f(x) = f(a) + (x − a) + (x − a)2 + ... + (x − a)n 1! 2! n! f (n+1)(c) + (x − c)n(x − a). n! (v) The series expansions for the elementary functions log(1 + h) and ex.

5. Functions of bounded variation 5.1. Properties of monotonic functions. (i) Let f be an increasing function defined on [a, b] and let and let P =

{x0, x1, . . . , xn} be a partition of [a, b]. Then we have

n−1 X + − [f(xk ) − f(xk )] ≤ f(b) − f(a) . n=1 (ii) If f is montonic on [a, b], then the set of discontinuities of f is countable. 6

5.2. Functions of bounded variation. (i) Functions of bounded variation with examples. (ii) If f is monotonic on [a, b], then f is of bounded variation on [a, b]. (iii) If f is continuous on [a, b] and if f 0 exists and is bounded in the interior, say |f 0(x)| ≤ A for all x ∈ (a, b), then f is bounded variation on [a, b]. P (iv) If f is of bounded varaiation on [a, b] and |∆fk| ≤ M for all partitions [a, b], then f is bounded on [a, b].

5.3. Total variation Vf (a, b) and its properties.

(i) Total variation Vf (a, b) of f in [a, b]. (ii) Suppose that f and g are each of bounded variation on [a, b]. Then so are their sum, difference, and product. (iii) Let f be of bounded variation on [a, b] and suppose that there exists a positive number m such that 0 < m ≤ |f(x)| for all x ∈ [a, b]. Then 1/f is also of bounded variation in [a, b]. (iv) Let f be of bounded variation on [a, b], and assume that c ∈ (a, b). Then f is of bounded varation on [a, c] and on [c, b] and we have

Vf (a, b) = Vf (a, c) + Vf (c, b)

. (v) Let f be of bounded variation on [a, b]. Let V be defined on [a, b] as

follows: V (x) = Vf (a, x) if a < x ≤ b, V (a) = 0. Then: (a) V is an increasing function on [a, b]. (b) V − f is an increasing function on [a, b].

5.4. Continuous functions of bounded variation.

(i) Let f be of bounded varation on [a, b] and let V (x) = Vf (a, x) if a < x ≤ b with V (a) = 0. Then every point of continuity of f is also a point of continuity of V . (ii) A countinous function f on [a, b] is of bounded varaion on [a, b] if and only if f can be expressed as the difference of two increasing continuous functions.

5.5. Rectifiable paths and arc length.

(i) Boundedness of the arc-length Λf (a, b) and rectifiability. n (ii) Let f = (f1, . . . , fn):[a, b] → R be a path. The f is rectifiable if and only if each component fk is of bounded varation on [a, b]. (iii) If c ∈ (a, b), then we have Λf (a, b) = Λf (a, c) + Λf (c, b). 7

(iv) Consider a rectifiable path f defined on [a, b]. If x ∈ (a, b], let s(x) =

Λf (a, x) and let s(a) = 0. Then we have: (a) The function s is icreasing and continuous on [a, b] (b) If there exists no subinterval of [a, b] on which f is constant, then s is strictly increasing on [a, b].

6. The Riemann-Stieltjes Integral 6.1. Linear Properties. (i) Riemann-Stieltjes integrability of f with respect to α on [a, b](f ∈ R(α)).

(ii) If f, g ∈ R(α) on [a, b], then c1f + c2g ∈ R(α) (for any 2 constants c1, c2 ∈ R), and we have Z b Z b Z b (c1f + c2g) dα = c1 f dα + c2 g dα. a a a

(iii) If f ∈ R(α) on [a, b] and f ∈ R(β) on [a, b], then f ∈ R(c1α + c2β) on [a, b], and we have Z b Z b Z b f d(c1α + c2β) = c1 f dα + c2 f dβ. a a a (iv) Assume c ∈ (a, b). If two out of the three integrals in the following equation exists, then the third also exists Z c Z b Z b f dα + f dα = f dα. a c a .

6.2. Integration by parts. (1) If f ∈ R(α) on [a, b], then α ∈ R(f) on [a, b], and we have Z b Z b f dα + α df = f(b)α(b) − f(a)α(a). a a 6.3. Change of variable. (i) Let f ∈ R(α) on [a, b] and let g be a strictly monotonic continuous function defined on the interval [c, d]. Assume that a = g(c) and b = g(d). Let h and β be the composite functions h = f ◦ g and β = α ◦ g on [c, d]. Then h ∈ R(β) on [c, d], and we have Z b Z d f dα = h dβ. a c 8

6.4. Reduction to Riemann integral. (i) Assume that f ∈ R(α) on [a, b] and assume that α has a continuous 0 R b 0 derivative α on [a, b]. Then the Riemann integral a f(x)α (x) dx exists, and we have

Z b Z b f(x)dα(x) = f(x)α0(x) dx a a .

6.5. Step functions as integrators. (i) Step function with examples. (ii) Given a < c <, define α on [a, b] as follows: The values of α(a), α(c), α(b) are arbitrary;

α(x) = α(a), if x ∈ [a, c) α(x) = α(b), if x ∈ (c, b].

Let f be defined on [a, b] in sucha way that at least one of the functions f or α is continuous from left or right at c. Then f ∈ R(α) on [a, b], and we have Z b f dα = f(c)[α(c+) − α(c−)]. a

(iii) Let P = {x0, x1, . . . , xn} ∈ P[a, b]. Let α be a step function defined on [a, b] with jump αk at xk. Let f be defined on [a, b] in such a way that not both f and α are discontinuous from the right or from the lest at R b each xk. Then a f dα exists and we have

n Z b X f dα = f(xk)αk. a n=1 (iv) Every finite sum can be written as a Riemann-Stieltjes integral. (v) Euler’s summation formula: If f has a continuous derivative f 0 on [a, b], then we have

n=[b] X Z b Z b f(n) = f(x) dx + f 0(x)((x)) dx + f(a)((a)) − f(b)((b)), n=[a]+1 a a

where ((x)) = x − [x]. 9

6.6. Monotonically increasing integrators: Upper and lower inte- grals. (i) Upper and lower Stieltjes sums of f with respect to α for the partition P : U(P, f, α) and L(p, f, α). Let α ↑ on [a, b]. Then (a) If P 0 ⊇ P , we have

U(P 0, f, α) ≤ U(P, f, α) and L(P 0, f, α) ≥ L(P, f, α).

(b) For any two partitions A aand B, we have

L(P1, f, α) ≤ U(P2, f, α).

R b (ii) Upper and lower Stieltjes integrals with respect to α: I(f, α) = a f dα R b and I(f, α) = a f dα. (iii) Riemann’s condition for integrability. (iv) Let α ↑ on [a, b]. Then the following statements are equivalent: (a) f ∈ R(α) on [a, b]. (b) f satisfies Riemann’s condition with respect to α on [a, b]. (c) I(f, α) = I(f, α).

6.7. Comparison Theorems. (i) Let α ↑ on [a, b]. If f, g ∈ R(α) on [a, b] and f ≤ g on [a, b], then we have Z b Z b f dα = g dα. a a (ii) Let α ↑ on [a, b]. If f ∈ R(α) on [a, b], then |f| ∈ R(α) on [a, b] and we have the inequality

Z b Z b

f dα ≤ |f| dα. a a (iii) Let α ↑ on [a, b]. If f ∈ R(α) on [a, b], then f 2 ∈ R(α) on [a, b].

6.8. Integrators of bounded variation. (i) Let α is of bounded variation on [a, b]. Let V (x) denote the total variation of α on [a, x] if x ∈ (a, b], and let V (a) = 0. Let f be defined and bounded on [a, b]. If f ∈ R(α) on [a, b], then f ∈ R(V ) on [a, b]. (ii) Let α be of bounded variation on [a, b] and assume that f ∈ R(α) on [a, b]. Then f ∈ R(α) on every subinterval [c, d] of [a, b]. 10

R x (iii) Let f, g ∈ R(α) on [a, b], where α ↑ on [a, b]. Define F (x) = a f dα R x and G(x) = a g dα, for all x ∈ [a, b]. Then fg ∈ R(α), and we have Z b Z b Z b fg dα = f dG = g dF. a a a

6.9. Necessary and Sufficient conditions for the existence of Riemann- Stieltjes integral. (i) If f is continuous on [a, b] and if α is of bounded variation on [a, b], then f ∈ R(α) on [a, b]. R b (ii) The Riemann integral a f(x) dx exists if any one of the following conditions hold. (a) f is continuous on [a, b]. (b) f is of bounded variation on [a, b]. (iii) Let α ↑ on [a, b] and let a < c < b. Assume further that both α and f R b are discontinuous from the right at c. Then the integtral a f dα does not exist.

6.10. Mean-Value Theorems for Riemann-Stieltjes integral. (i) First Mean-Value Theorem: Let α ↑ and let f ∈ R(α) on [a, b]. Let M and m denote the sup and inf of the set {f(x): x ∈ [a, b]}. Then there exists c ∈ [m, M] such that

Z b Z b f dα = c dα(x) = c(α(b) − α(a)). a a (ii) Second Mean-Value Theorem: Let α be continuous and f ↑ on [a, b].

Then there exists a point x0 ∈ [a, b] such that

Z b Z x0 Z b f dα = f(a) dα(x) + f(b) dα(x). a a x0

6.11. Integral as a function of the interval. (i) Let α be of bounded variation on [a, b] and assume that f ∈ R(α) on R x [a, b]. Define F by F (x) = a f dα, if x ∈ [a, b]. Then we have: (a) F if of bounded variation on [a, b]. (b) Every point of continuity of α is also point of continuity of F . (c) If α ↑ on [a, b], then the derivative F 0(x) exists at each point x ∈ (a, b) where α0(x) exists and where f is continuous. For such x, we have F 0(x) = f(x)α0(x). 11

(ii) If f, g ∈ R on [a, b]. Then F and G are continuous functions of bounded variation on [a, b]. Also, f ∈ R(G) and g ∈ R(F ) on [a, b], and we have Z b Z b Z b f(x)g(x) dx = f(x) dG(x) = g(x) dF (x). a a a (iii) Second fundamental theorem of integral calculus: Let f ∈ R on [a, b]. Let g be a function on [a, b]. Let g be a function defined on [a, b] such that the derivative g0 exists in (a, b) and has the value g0(x) = f(x), for every x ∈ (a, b). At the end points assume that g(a+) and g(b−) and satisfy g(a) − g(a+) = g(b) − g(b−). Then we have Z b f(x) dx = g(b) = g(a). a 6.12. Change of variable and Riemann integral. (i) Let g0 be continuous on [c, d]. Let f be continuous on g([c, d]) and R x define F by the equation F (x) = g(c) f(t) dt, if x ∈ g([c, d]). Then for each x ∈ [c, d] the integral f ◦ g exists and has the value F [g(x)].

6.13. Riemann-Stieltjes integrating depending on a parameter. (i) Let f be a continuous at each point in (x, y) ∈ [a, b] × [c, d]. Assume that α is of bounded variation on [a, b] and let F be the function defined R b on [c, d] by f(y) = a f(x, y) dα(x). Then F is continuous on [c, d]. (ii) If f is continuous in the rectangle [a, b] × [c, d], and if g ∈ R on [a, b], R b then the F (x) = a g(x)f(x, y) dx is continuous on [c, d]. 6.14. Differentiation under the integral sign. (i) Let α be of bounded variation on [a, b], and for each y ∈ [c, d], assume R b that f(y) = a f(x, y) dα(x) exists. If the partial derivative D2f is continuous on [a, b] × [c, d]. then the derivative F 0(y) exists for each y ∈ (c, d), and is given by Z b 0 F (y) = D2f(x, y) dα(x). a 6.15. Interchanging the order of integration. (i) Let α be of bounded variation on [a, b], β be of bounded variation on [c, d], and let f be continuous on [a, b] × [c, d]. For (x, y) ∈ [a, b] × [c, d], R b R b define F (y) = a f(x, y) dα(x) and G(x) = a f(x, y) dβ(x). Then F ∈ R(β) on [c, d], g ∈ R(α) on [a, b], and we have Z d Z b F (y) dβ(y) = G(x) dα(x). c a 12

6.16. Lebesque’s criterion. (i) Set of measure zero. (ii) A countable union of sets of measure zero is also of measure zero (iii) Lebesque’s criterion for Riemann-integrability: Let f be defined and bounded on [a, b], and let D denote the set of discontinuities of f in [a, b]. Then f ∈ R on [a, b] if and only if D has measure zero.

7. Introduction to topology 7.1. Basic notions. (i) Topological spaces with examples. (ii) Closed and open sets. (iii) Closure of a set and dense sets. (iv) Basis for a topology. (v) Metric topology. (vi) Continuous functions.

7.2. Compactness. (i) Compact spaces with examples. (ii) Continuous image of a compact space is compact. (iii) Extreme Value Thorem: Continuous real-valued function on a compact sets attains its maximum and minimum. (iv) Continuous function on a compact set is uniformly continuous. (v) Heine-Borel Theorem: Compact subset of a metric space is closed and bounded. n (vi) A closed and bounded subset of R is compact. (vii) Let K be a compact subset of a metric space (X, d). Then the following are equivalent: (a) K is compact. (b) K is sequentially compact. (c) K is complete and totally bounded.