(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define

Transcription

1 Homework, Real Analysis I, Fall, (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that (S, ρ) is a metric space. (Hint: The continuity of the elements of S is crucial.) (2) Consider the extended real line [, ] with the standard topology as defined in Rudin. Let f : (, ) [, ] and let x f 1 ( ). Show that f is continuous at x if and only if the following holds: For each N > 0 there is a δ > 0 so that y x < δ = f(y) > N. (3) Let be a set, and consider F to be the set of subsets of consisting of single points. In other words, F = {{x} x }. (a) Find an explicit description of all the elements of the σ- algebra M generated by F. (b) Describe, with proof, all sets satisfying M = 2. (You may use the following fact: Each uncountable set has an uncountable subset Y so that Y c is also uncountable.) (4) Rudin, chapter 1, problem 4. (5) Let be a set, b k : [, ] for k = 1, 2, 3,..., and h = inf k b k. Show that h 1 ((α, ]) = m=1 k=1 b 1 k ((α + 1, ]). m (6) Rudin, chapter 1, problem 5. (7) Rudin, chapter 1, problem 10. (8) Rudin, chapter 1, problem 12. (Hint: Prove the statement by contradiction: So there is an ɛ > 0 and measurable sets E n for each natural number n with µ(e n ) < 2 n but E n f dµ ɛ. Then apply Fatou s Lemma to the functions { f (1 χ En )}. The key point is to evaluate lim inf n f (1 χ E n ) dµ = f dµ lim sup f χ En dµ. n Do this by showing lim sup χ En = χf for a measurable set F of measure 0.) 1

2 (9) Let f : (, ) [, ] be a function whose restriction to (, 0) (0, ) is continuous. Assume the one-sided limits p = lim x 0 f(x), q = lim x 0 + f(x) exist as extended real numbers. Show that f is upper semicontinuous on (, ) if and only if f(0) max{p, q}. (10) Let (, d) be a compact metric space which contains at least 2 points. Consider a point p. Show that {p} is compact if and only if inf{d(p, x) x p} > 0. (11) Let be a Hausdorff topological space, and let K, K be two disjoint compact subsets of. Prove that there are disjoint open sets U, U so that K U and K U. (Hint: Use Theorem 2.5 and mimic its proof.) (12) Rudin, chapter 2, problem 1. (Hint: to find a counterexample, consider h n (x) = χ (,0) +χ [0,1] (1 x n ). Compute lim n h n (x). Let f 1 = h 1 and f n = h n h n 1. Is the sum i=1 f i upper semicontinuous? lower semicontinuous?) (13) Rudin, chapter 2, problem 4. (Hint for 4b: Use the fact that E is inner regular to find a large compact subset K 1 of E. Then repeat the process to find a large compact subset K 2 of E K 1. Show that if the compact sets K i are chosen so that the measures of E K 1, E K 1 K 2, etc. are small enough, then E (K 1 K 2 ) has measure 0.) (14) Let be a topological space and M be a σ-algebra on which contains all Borel sets. Let m, µ be two positive measures on M. Assume there is a constant c [0, ) so that µ(e) = c m(e) for each Borel set E. Assume that m is regular. Let A M and m(a) = 0. Prove µ(a) = 0. (15) (Not assigned) Let m denote Lebesgue measure on R k. (a) Show that the sum 2 n1 nk <, where n = (n 1,..., n k ). n Z k (Hint: use the corollary on p. 23 of Rudin to evaluate the sum.) (b) Let x = (x 1,..., x k ) be coordinates in R k. Let P be the hyperplane (i.e., linear subspace of dimension k 1) given by P = {x : x k = 0}. Let ɛ > 0. Construct an open set V P so that m(v ) < ɛ. (Hint: Write V as a countable 2

3 union of open rectangular k-cells. It may help to do the case k = 2 first.) (c) Show that m(p ) = 0. (d) Let L R k be any linear subspace of dimension k 1. Show m(l) = 0. [Hint: Show there is an invertible linear map T : R k R k so that T (P ) = L. Then use Problem 14, and Theorem 2.20e for T invertible (which is the case we proved in class).] (e) If J R k is any linear subspace of dimension l < k, show that m(j) = 0. (Hint: Show that such a J is always a subset of a subspace L of the form considered in part d.) (16) The Cantor set. (a) Define subsets of the unit interval O n, n inductively by O 1 = ( 1 3, 2 3 ), O n+1 = 1 3 O n ( 1 3 O n ), 0 = [0, 1], n+1 = n O n. Show that n is a disjoint union of closed intervals of each of length 3 n. Hint: Show the following inductive definition for n holds: 0 = [0, 1], n+1 = 1 3 n ( 1 3 n ). (b) Define the Cantor set K = ( ) n = 0 O n. n=1 Show K is a compact subset of R with Lebesgue measure 0. (c) Show that K consists of all numbers in the unit interval [0, 1] which have a base-3 expansion which only uses 0 s and 2 s. (One special point to recall is that (0. 2) 3 = (1. 0) 3, where the bar signifies infinite repetition of the ternary digits. Therefore, the endpoint 1 is legitimately in K.) (d) Construct a surjective map from K onto [0, 1]. (Hint: Relate the base-3 expansions of elements of K to base-2 expansions of elements of [0, 1].) Conclude that K is uncountable. n=1 3

4 (17) Let f be an L 1 function with respect to Lebesgue measure on R. Use the result of Chapter 1, Problem 12, to show that g(x) = x 0 f(t) dt is continuous. (18) (Not assigned) Rudin, chapter 3, problem 3 (19) Rudin, chapter 3, problem 4 (Hint: The key to this problem is to use Hölder s Inequality to compute bounds on f p p = f p dµ = f p γ f γ dµ. Then vary the different parameters γ and the one from Hölder s Inequality to prove new inequalities among the different L p norms. Examples of functions to try for part (c) are x l and x l log x m near both 0 and (so multiply them by characteristic functions such as χ (0, 1 2 ] and χ [2, ) to try to construct examples). For part (e), it suffices to prove lim inf f p f, lim sup f p f. p p For the first inequality, use the definition of f. For the second, you should use Hölder s inequality as above to relate f p to f r < and f.) (20) Rudin, chapter 3, problem 10 (21) (Not assigned) Rudin, chapter 3, problem 11 (22) Rudin, chapter 3, problem 12 (Hint: Use Jensen s Inequality.) (23) (Not assigned) Rudin, chapter 3, problem 13 (24) The p-adic numbers. Let p {2, 3, 5, 7, 11,... } be a prime number. (a) For x Q, define { 0 for x = 0, x p = p n for x = p n (a/b), where a, b are not multiples of p. Show that x y p is a metric on Q. (b) Define the p-adic numbers Q p to be the completion of Q with respect to the metric x y p. Let k be an integer. Assume a j {0, 1,..., p 1} for j k. Then show that a j p j j=k 4

5 converges in Q p. (In other words, show the partial sums of the series are a Cauchy sequence in Q with respect to the p-adic metric.) (c) Let { l } T = a j p j k, l Z, k l, aj {0, 1,..., p 1}. j=k Show that T = {sp n s, n are nonnegative integers}. (d) Show that the completion of T with respect to the p-adic metric x y p may be identified with the set { } = a j p j k Z, aj {0,..., p 1}. j=k What is the p-adic metric on? (e) Show that if x, y that x + y and xy. (Define x+y and xy by addition and multiplication of power series, with the usual carrying operation from decimal arithmetic to handle the case of coefficients p.) (f) Show that 1 = (p 1)[1 + p + p 2 + ]. (g) Define the set Z p to consist of all elements of for which a j = 0 for all j < 0. (Elements of Z p are called the p-adic integers.) Show that Z p = {x x p 1}. (h) If x j Z p for all j k, show that the sum x j p j j=k converges in. (i) If q {1,, p 1}, show there is a unique r {1,..., p 1} so that qr = 1 + ap for some a {0,, p 1}. (This is just the usual fact that Z/pZ is a field.) (j) Show that the multiplicative inverse of 1 + ap exists in and is given by 1 ap + a 2 p 2 a 3 p 3 +. (k) Show that any rational number can be represented by a unique element of. (Hints: Write any rational number as a product of ±1, an element of T, and a reciprocal q 1 of a positive integer q which is not divisible by p. It will also be useful to know that b Z p implies b Z p. To show uniqueness, analyze the power series term by term to ensure the relevant equations are satisfied: For example, we 5

6 must have for each rational number t that rational number t satisfies t + ( t) = 0.) (l) Show that is the completion of Q with respect to the p-adic metric (and so we may identify with Q p ). (Hint: Show that for each rational number, the two definitions of p, on Q and on, agree.) (m) Show that Q p is a field. (So in addition, you must show that nonzero elements have multiplicative inverses.) (n) Show that every real number can be represented as a (not quite unique) sum ± k j= a j p j, k Z, a j {0,..., p 1}, which is convergent with respect to the usual metric x y on R. (25) (Not assigned) Equip = {x 1, x 2 } with the discrete topology and counting measure, and consider the space S of all realvalued functions from R. Identify S with R 2. Let p represent the L p norm as usual. (a) Sketch the region {f S : f p 1} for p = 1, 2,. Also sketch the region for representative values of p in (1, 2) and (2, ). (b) Show that the norm p on R-valued functions on comes from a real inner product if and only if p = 2. (The same thing holds for C-valued functions.) (26) (Not assigned) Let f : (a, b) R be a function. Show that f is convex if and only if the region above the graph {(x, y) x (a, b), y > f(x)} R 2 is convex. (27) Rudin, chapter 4, problem 2 (28) (Not assigned) Rudin, chapter 4, problem 5 (29) (Not assigned) Rudin, chapter 4, problem 6 (30) Rudin, chapter 4, problem 9 (31) Nonprincipal Ultrafilters (a) Let N = {1, 2, 3,... } be the set of natural numbers. filter on N is a nonempty subset ω 2 N satisfying: (i) / ω. (ii) A, B ω = A B ω. (iii) A ω, B A = B ω. Let Z = {A N A c < }. Show that Z is a filter. 6 A

7 (b) Let A N be a nonempty set. Show that {B : A B} is a filter. (c) Let F be a filter on N, and let A N. Show that at least one of the sets B = A or B = A c satisfies the property ( ) B P for all P F (d) Show that if F a filter and B is a subset of N which satisfy property (*), then G = G(F, B) = {Y there is a P F so that Y B P } is a filter containing F. (e) Use the Hausdorff Maximality Principle to show that there exists a maximal filter ω 2 N which contains any fixed filter F, where maximality is measured with respect to set inclusion. (f) Show that any maximal filter ω satisfies the additional condition that for all A N, ω contains exactly one of A and A c. Any maximal filter which contains Z is called a nonprincipal ultrafilter. (g) Show that a subset ω of 2 N is a nonprincipal ultrafilter if and only if (i) A < = A / ω. (ii) A, B ω = A B ω. (iii) A ω, B A = B ω. (iv) A N = exactly one of A, A c is in ω. (h) For F 2 N, consider the characteristic function µ = χ F : 2 N {0, 1}. Show that F is a nonprincipal ultrafilter if and only if µ is a finitely additive measure on N for which all finite sets have measure 0 and N has measure 1. (A finitely additive measure m on N is a function µ: 2 N [0, ] which satisfies µ( ) = 0, A B = = µ(a B) = µ(a) + µ(b). Note we do not assume µ is countably additive.) (i) Let (, d) be a metric space, and let ω 2 N be a nonprincipal ultrafilter. For x n a sequence in and x, we say x is the ω-limit of x n if for all ɛ > 0 In this case, we write {n d(x n, x) < ɛ} ω. x = lim ω x n. 7

8 Show that lim x n = x = x = lim x n. n ω (j) Show that if ω is a nonprincipal ultrafilter and x and y are two ω-limits of x n, then x = y. (k) Let (, d) be compact, and fix a nonprincipal ultrafilter ω. Show that every sequence x n has an ω-limit. (Hint: To prove the statement by contradiction, construct an open cover.) (32) (Not assigned) Let x n be a sequence in a Hilbert space H which converges strongly to x (i.e., x n x H 0 as n ). Use the Schwarz Inequality to show that x n converges to x weakly (i.e., that for all w H, (x n, w) (x, w) as n ). (33) Urysohn Lemma for C functions in R k. Let K V R k, where K is compact and V is open. (a) Show that there is an ɛ > 0 so that K ɛ = K + B ɛ (0) = {x + y x K, y Bɛ (0)} is contained in V. (b) Show that each K ɛ is compact (realize K ɛ as the image of the set K B ɛ (0) R k R k by a continuous map). (c) If f = χ Kɛ for ɛ as in part (a), and φ is a nonnegative C function on R k with compact support and integral one, and φ δ (x) = δ k φ(x/δ), show that for δ small enough, the convolution f φ δ is a C function satisfying K f φ δ V. 8