Written by M.van den Berg School of Mathematics University of Bristol BS8 1TW Bristol, UK

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Exercises. 1. Let X be a non-empty set, and suppose ρ : X × X → R satisfies 0 ≤ ρ(x, y) < ∞ for all x, y ∈ X, ρ(x, y) = 0 if and only if x = y, and ρ(x, y) ≤ ρ(x, z) + ρ(y, z) for all x, y, z ∈ X. Prove that ρ is a metric on X. 2. Let (X, d) be a metric space, let 0 < α ≤ 1 and let e(x, y) = d(x, y)α , x ∈ X, y ∈ X. Prove that e is a metric on X. 3. Prove that if (X, k · k) is a normed space then X is a metric space with metric ∀x, y ∈ X d(x, y) = kx − yk.

4. Let (X, d) be a metric space. n X d(xi−1 , xi ) for all x0 , x1 , . . . , xn ∈ X. Prove that d(x0 , xn ) ≤ i=1

5. Let (X1 , d1 ), (X2 , d2 ) be metric spaces. Let X = X1 × X2 . Define d : X 2 → R by ∀(x1 , x2 ), (y1 , y2 ) ∈ X q d((x1 , x2 ), (y1 , y2 )) = d21 (x1 , y1 ) + d22 (x2 , y2 ). Verify the axioms of the metric space. 6. Let B[0, 1] be the set of bounded functions on [0, 1] equipped with the supremum metric. Let (fn ) be the sequence in B[0, 1] defined by 1 − nx , 0 ≤ x ≤ n1 fn (x) = 0 , n1 ≤ x ≤ 1. (i) Compute d(fn , fm ), where n, m ∈ N, n ≥ m. (ii) Prove that (fn ) does not converge in {B[0, 1], d}. 7. Let C[0, 1] be the set of continuous real valued functions on [0, 1]. Define for f, g ∈ C[0, 1], Z 1 1/2 2 d(f, g) = |f (x) − g(x)| dx . 0

(i) Prove that d is a metric on C[0, 1]. (ii) Let e denote the supremum metric on C[0, 1]. Prove that fn → f in {C[0, 1], e} implies fn → f in {C[0, 1], d}.

8. Let S be the space of sequences x = (x1 , x2 . . . ) of real numbers such that absolutely. For x, y ∈ S put ∞ X ρ(x, y) = |xn − yn |. n=1

Prove that ρ is a metric on S.

P

xn converges

9. Let X be the set of all bounded sequences of real numbers: every element in X can be written as x = (ξ1 , ξ2 , . . . ) and there exists cx ∈ R such that |ξj | ≤ cx for all j. Define d(x, y) = supj∈N |ξj − ηj |, where x = (ξj ) = (ξ1 , ξ2 , . . . ) ∈ X, y = (ηj ) = (η1 , η2 , . . . ) ∈ X. Prove that d is a metric on X. 10. Find int(E), E, b(E) and E 0 in each of the following subsets of R2 with the euclidean metric: (i) E = {(x, y) : x2 + y 2 < 5} (ii) E = {(x, y) : 1 ≤ x2 + y 2 < 5} (iii) E = Q × {0} (iv) E = Q × (R\Q) 1 (v) E = : n ∈ N × Z. n

11. Let (X, d) be a metric space, A closed subset, x 6∈ A. Prove that d(x, A) := inf{d(x, y) : y ∈ A} > 0.

12. Prove that in Rn with the Euclidean metric any collection of disjoint open sets is at most countable. Is this true for an arbitrary metric space? 13.

(i) Prove that E is closed if and only if b(E) ⊂ E. (ii) Prove that b(F ) = ∅ if and only if F is open and closed.

14.

(i) Prove that int(E) is the union of all open sets contained in E (and so is the ‘largest’ open set contained in E). (ii) Prove that E is the intersection of all closed sets containing E (and so is the ‘smallest’ closed set containing E).

15.

(i) Prove that E1 ∪ · · · ∪ Em = E 1 ∪ · · · ∪ E m but that for an infinite collection of sets Ei , i ∈ I [ [ Ei ⊃ Ei. i∈I

(ii) Prove that

\ i∈I

Ei ⊂

\

i∈I

E i . Construct an example to show that equality need not hold

i∈I

even for a finite collection. 16.

(i) Prove that X\E = int (X\E). (ii) Prove that int(E) ∪ b(E) ∪ int(X\E) = X.

17. Let B[0, 1] have the supremum metric d, and let f ∈ B[0, 1] be given by f (t) = t2 , 0 ≤ t ≤ 1. (i) Describe the open ball B(f ; 1). ˜ ; 1). (ii) Describe the closed ball B(f ˜ ; r) contains g, g(t) = t(1 − t). (iii) Find the least value of r such that B(f 2

(iv) Let C[0, 1] ⊂ B[0, 1] be the set of continuous functions from [0, 1] into R. int(C[0, 1]) and b(C[0, 1]).

Find

18. A set A in a metric space is bounded if the diameter of A is finite: i.e. diam(A) = sup{d(a1 , a2 ) : a1 ∈ A, a2 ∈ A} < ∞. (i) Prove that if diam(A) < ∞, then diam(A) = diam(A). (ii) Prove that if B and C are bounded sets with B ∩ C 6= ∅ then diam(B ∪ C) ≤ diam(B) + diam(C). (iii) Prove that if A1P , . . . An are bounded sets with Ai−1 ∩ Ai 6= ∅ for i = 2, . . . , n then n diam(∪ni=1 Ai ) ≤ i=1 diam(Ai ). (iv) Give an example of bounded sets E and F such that diam (E∪F ) > diam(E)+diam(F ).

19. Let (xi ) be a sequence of distinct elements in a metric space, and suppose that xi → x. Let f be a one-to-one map of the set of xi ’s into itself. Prove that f (xi ) → x. 20. Let (X, d) be a metric space. A set D is said to be dense in E if D ⊂ E ⊂ D. Show that if C is dense in D, and D is dense in E, then C is dense in E. 21. Let B[a, b] have the supremum metric d. Let E = {f ∈ B[a, b] : f (x) > 0, a ≤ x ≤ b}, F = {f ∈ C[a, b] : f (x) > 0, a ≤ x ≤ b}. (i) Describe int(E), E, and b(E). (ii) Describe int(F ), F , and b(F ).

22. Let f : (0, ∞) × (0, ∞) → [0, ∞) be given by f ((x, y)) = Find all α, β such that

lim

xα y β . x+y

f ((x, y)) exists.

(x,y)→(0,0)

23. Let (X, ρ), (Y, σ) be metric spaces. Let f, g : X → Y be continuous. Prove that the set B = {x ∈ X : f (x) = g(x)} is a closed subset of X. 24. Let (X, ρ) and (X, σ) be metric spaces, and let I : (X, ρ) → (X, σ) be the identity function given by I(x) = x, x ∈ X. (i) Prove that I is a bijection. (ii) Prove that both I and I −1 are continuous if and only if ρ and σ are equivalent metrics. (iii) Give an example to show that when ρ and σ are not equivalent I may be continuous without I −1 being continuous.

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25. Let A be a non-empty set in a metric space (X, d). Define f : X → R by f (x) = inf{d(a, x) : a ∈ A}. Prove that f is continuous. 26. Let (X, ρ), (Y, σ) and (Z, τ ) be metric spaces, and let f : X → Y , g : Y → Z be continuous functions. Prove that g ◦ f : X → Z is continuous. 27. Let (X, ρ) and (Y, σ) be metric spaces and let f and g be continuous functions from X into Y which are equal on a dense subset of X. Prove that f = g on all of X. 28. Let (X, ρ) and (Y, σ) be metric spaces. Prove that f : X → Y is continuous if and only if f (H) ⊂ f (H) for all sets H in X. 29. Which of the following subsets of R2 with the Euclidean metric are complete: (i) {(x, y) : x2 + y 2 < 2}, (ii) Z × Z, (iii) (x, y) : x > 0, y ≥ x1 , (iv) {(x, y) : 0 < x < 1, y = 0}? 30. Define d : R × R → R by d(x, y) = |arctanx − arctany|. (i) Prove that (R, d) is a metric space. (ii) Prove that the sequence (xn ) with xn = n is a Cauchy sequence in (R, d) which does not converge.

31.

(i) Prove that the intersection of any collection of complete subsets of a metric space is complete. (ii) Prove that the union of a finite number of complete subsets of a metric space is complete.

32. Let X be the set of all ordered n-tuples x = (x1 , . . . , xn ) of real numbers, and define d : X × X → R by d(x, y) = max |xj − yj |, where y = (y1 , . . . , yn ). j

(i) Prove that (X, d) is a metric space. (ii) Prove that X is complete.

33. Let (X, d) and (Y, e) be complete metric spaces, and let M = X × Y . Define f : M × M → R by f ((x1 , y1 ), (x2 , y2 )) = d(x1 , x2 ) + e(y1 , y2 ). (i) Prove that (M, f ) is a metric space. (ii) Prove that M is complete.

x y 34. Define σ : R × R → R by σ(x, y) = − . 1 + |x| 1 + |y| (i) Prove that (R, σ) is a metric space. (ii) Prove that R is not complete. (iii) Prove that σ is equivalent to the usual metric on R. 4

(iv) The metrics ρ, σ on a set X are equivalent. Are (X, ρ) and (X, σ) necessarily both complete or both not complete?

35.

(i) Prove that if every closed ball of a metric space (X, d) is complete then X is complete. (ii) Prove that if every countable closed subset of a metric space (X, d) is complete then X is complete.

36. Let T : [1, ∞) → [1, ∞) be defined by Tx = x +

1 . x

Is T a contraction? Does it have a fixed point? 37. Let T : [−1, 1] → R be given by T x = 1 − cx2 , where R has the usual metric. (i) Find all values of c such that T is a contraction mapping from [−1, 1] into [−1, 1]. (ii) Let c =

1 3

and x0 = 0. Compute the iterates T j x0 for j = 1, 2, 3 and find lim T j x0 . j→∞

38. Let x∗ be the positive real root of 2 sin x = x. (i) Find a closed interval X in R containing x∗ such that T : X → X given by T x = 2 sin x is a contraction mapping. (ii) Use the contraction mapping theorem and a calculator to find an approximation for x∗ within 10−4 . π (iii) Suppose x0 = . Find an integer N such that |T n x0 − x∗ | < 10−12 for n > N . 2 1

Z

xyψ(y)dy + x2 ,

39. Consider the integral equation ψ(x) = λ

−1 ≤ x ≤ 1.

−1

(i) Prove that there exists a unique solution ψ ∈ C[−1, 1] (the real valued continuous functions on [−1, 1]) for |λ| < 21 . (ii) Find the solution of the integral equation for |λ| < 21 . What happens if λ = 32 ? dy = x + y 2 with initial condition y(0) = 0. dx

40. Consider the differential equation

(i) Prove that there exists a unique solution on [−α, α] for any α ∈ (0, 2−2/3 ). (ii) Find the first few terms of the solution the solution about 0.

41. Let C[0, 1] have the supremum metric d, and let T : C[0, 1] → C[0, 1] be defined by Z T f (t) =

t

f (τ )dτ,

0 ≤ t ≤ 1,

f ∈ C[0, 1].

0

Prove that T is not a contraction mapping but that T 2 is a contraction mapping.

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42. Let (X, d) be a complete metric space, and let Ti , i = 1, 2, . . . and T be contraction mappings on X with fixed points xi , i = 1, 2 . . . and x respectively. Prove that if Ti converges uniformly to T then limi→∞ xi = x. 43. Let (X, d) be a complete metric space, and let T : X → X. Suppose T 2 = T ◦ T is a contraction mapping. Prove that T has a unique fixed point p in X, and that for any x ∈ X, limj→∞ T j x = p. 44. Let A1 , . . . , An be compact sets in a metric space (X, d). Prove that A1 ∪ · · · ∪ An is compact. 45. Let E be a compact set in a metric space (X, d). Let diam(E) denote the diameter of E. (i) Prove that there exist points x, y ∈ E such that diam(E) = d(x, y). (ii) Give an example to show that these points are not necessarily unique.

46. Which of the following subsets of R2 with the usual metric are compact? (i) {(x, y) ∈ R2 : x = 0}, (ii) {(x, y) ∈ R2 : x2 + y 2 = 1}, (iii) {(x, y) ∈ Z2 : x2 + y 2 < 25}, (iv) {(x, y) ∈ R2 : x ≤ y}, (v) {(x, y) ∈ R2 : x2 + y 2 = 1, x ∈ Q}. (vi) {(x, y) ∈ R2 : y = sin x1 , 0 < x ≤ 1} ∪ {(x, y) ∈ R2 : −1 ≤ y ≤ 1, x = 0}. 47. Let T be a compact metric space with metric d, and suppose that f : T → T is a continuous map such that for every x ∈ T , f (x) 6= x. Prove that there exists an > 0 such that d(f (x), x) ≥ for all x ∈ T . (Hint: consider g : T → R given by g(x) = d(f (x), x)). 48. Determine whether f : A → R is uniformly continuous on A, when (i) A = (0, 1), f (x) = x, (ii) A = (0, 1), f (x) = 1/(1 − x), (iii) A = R, f (x) = cos x, √ (iv) A = (0, 1), f (x) = sin(1/x), (v) A = [0, ∞), f (x) = x, (vi) A = [1, ∞), f (x) = 1/x. 49. Let f : [0, ∞) → R be continuous, and suppose that limx→∞ f (x) exists. Prove that f is uniformly continuous. 50. The C[0, 1] with the supremum metric are given by f (φ)(x) = R x functions f, g : C[0, R x 1] → 2 φ(t)dt, g(φ)(x) = φ(t) dt (φ ∈ C[0, 1], x ∈ [0, 1]). Show that both functions are con0 0 tinuous on C[0, 1]. Are they uniformly continuous? 51. Let B[0, 1] be the set of bounded real valued functions on [0, 1] with the usual metric 1 d(f, g) = sup |f (x) − g(x)|. Let fn : [0, 1] → R be given by fn (x) = 1 if x = , fn (x) = 0 n 0≤x≤1 1 if x 6= . n 6

(i) Prove that fn ∈ B[0, 1]. (ii) Prove that (fn ) does not have a convergent subsequence. ˜ (iii) Let φ ∈ B[0, 1] be defined by φ(x) = 0, 0 ≤ x ≤ 1. Prove that B(φ; 1) is closed, bounded, and complete but not compact.

52. Give an example of a metric space (X, d) with disjoint, closed subsets A, B such that inf{d(x, y) : x ∈ A, y ∈ B} = 0. 53. Let (X, d) be a metric space with disjoint compact subsets A, B. Prove that there exist a ∈ A, b ∈ B such that inf{d(x, y) : x ∈ A, y ∈ B} = d(a, b), and conclude that d(a, b) > 0. 54. A real-valued map f on a metric space (X, d) is lower semi-continuous if for every a ∈ R, f −1 (a, ∞) is open in X. Prove that if X is compact and f : X → R is lower semicontinuous then f is bounded below and attains its lower bound on X. 55. Which of the following subsets of R2 with the usual metric are connected? (i) B((0, 0); 1) ∪ B((2, 0); 1), ˜ (ii) B((0, 0); 1) ∪ B((2, 0); 1), ˜ ˜ (iii) B((0, 0); 1) ∪ B((2, 0); 1), (iv) the set of all points with at least one coordinate in Q.

56. Prove that the components of a closed subset of a metric space are closed. 57. Let f : (0, ∞) → R2 be given by f (x) = (x, sin x1 ). (i) Prove that f is continuous. (ii) Prove that f (0, ∞) is connected. (iii) Prove that f (0, ∞) ∪ {(x, y) ∈ R2 : x = 0, −1 ≤ y ≤ 1} is connected.

58. Prove that the intersection of two connected sets in R is connected. Show by an example that this is false in R2 . 59. Let Q be the rationals with metric d(q1 , q2 ) = |q1 − q2 |. Prove that the only non-empty connected sets are those consisting of single points. 60. Show by means of an example that the components of an open set are not necessarily open. 61. Give an example of closed connected subsets Cn ⊂ R2 such that Cn ⊃ Cn+1 for all ∞ \ n = 1, 2, . . ., but Cn is not connected. n=1

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62. Prove that C[0, 1] with the supremum metric is connected. 63. Let S = {(x1 , x2 ) ∈ R2 : x21 + x22 = 1}, and let f : S → R be a continuous map. Show that there exists a point x = (x1 , x2 ) ∈ S such that f (x) = f (−x), where −x = (−x1 , −x2 ). (Hint: consider g : S → R given by g(x) = f (x) − f (−x).) 64. Which of the following sequences (fn ) converge uniformly on [0, 1]? (i) fn (x) =

x , 1 + nx

2

(ii) fn (x) = nxe−nx , 2

(iv) fn (x) = nx(1 − x2 )n , (v) fn (x) =

1

(iii) fn (x) = n 2 x(1 − x)n ,

xn , (vi) fn (x) = n−x xn cos(nx). 1 + xn

65. The set X consists of the real functions φ on [a, b] such that on [a, b] φ0 exists and is continuous, and |φ(x)| ≤ M , |φ0 (x)| ≤ M . For n = 1, 2, . . . the functions fn : X → R Z b are defined by fn (φ) = φ(x) sin(nx)dx, φ ∈ X. Prove that the sequence (fn ) converges a

uniformly on X.

66. The functions fn on [0, 1] are given by fn (x) =

nx 1 + n2 xp

(p > 0).

(i) For what values of p does the sequence converge uniformly to its pointwise limit f ? Z 1 Z 1 (ii) Examine whether fn (x)dx → f (x)dx for p = 2, p = 4. 0

0

67. The functions fn on [−1, 1] are given by fn (x) =

x . 1 + n2 x2

(i) Show that (fn ) converges uniformly and that the limit function f is differentiable. (ii) Show that the relation f 0 (x) = lim fn0 (x) does not hold for all x in [−1, 1]. n→∞

68. Let (fn ) be a sequence of functions from [a, b] into R. Prove that if (fn ) converges pointwise on [a, b] to a continuous function f , and that if each fn is monotonically increasing, then the convergence is uniform. 69. Let φ ∈ C[0, 1] and let fn (x) = xn φ(x), 0 ≤ x ≤ 1. Prove that (fn ) converges uniformly on [0, 1] if and only if φ(1) = 0. 70. Let fn (x) = (1 + xn )1/n , 0 ≤ x ≤ 2. Prove that the sequence (fn ) of differentiable functions converges uniformly to a limit function which is not differentiable at x = 1. 71. Let fn (x) = (sin(nπx))n ,

0 ≤ x ≤ 1. 8

(i) Examine whether limn→∞ fn exists . R1 (ii) Examine whether limn→∞ 0 fn (x)dx exists.

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