Take-Home Final Examination of Math 55a (January 17 to January 23, 2004)

Transcription

1 Take-Home Final Eamination of Math 55a January 17 to January 3, 004) N.B. For problems which are similar to those on the homework assignments, complete self-contained solutions are required and homework problems cannot be quoted simply as known facts in the solutions. Notations. N = all positive integers. Z = all integers. R = all real numbers. C = all comple integers. F means either C or R. Problem 1. The five aioms of Peano are the following. 1) The set N of all natural numbers contains an element 1. ) There is an immediate successor N defined for every element N. 3) 1 is not an immediate successor of any element of N. 4) Two distinct elements of N have distinct immediate successors. 5) If a subset E of N contains 1 and contains the immediate successor of every one of its elements, then E must be all of N. Addition in N is defined by +1 = and +y = + y). Multiplication in N is defined by 1 = and y = y)+. From the five Peano s aioms and the definitions of addition and multiplication prove that y + z) = y) + z) for,y,z N. Problem. For every natural number N let X be a nonempty) metric space with metric d X, ). Let X be the product space N X. We denote the components of an element X by so that we write = { } =1 with X. Let ρ > 1. Define the metric d X, ) on X by { for 1) = 1) d X 1), )) = } =1 { and ) = =1 ) 1 ρ } 1 d X 1), ) ) 1 + d X 1), ) =1. ).

2 a) Verify that d X is indeed a metric on X. b) Verify that { a} subset G of X is open in X if and only if for every point 0) = of G there eist some N N and some positive 0) =1 numbers r 1, ) r,, r N such that every point = { } =1 of X with d X, 0) < r for 1 N belongs to G. c) Show that X is compact if and only if each X is compact. Problem 3. Let X and Y be metric spaces and f : X Y be a surjective continuous map. Assume the following three conditions. a) X is compact. b) f 1 y) is connected for every y Y. c) Y is connected. Prove that X is connected. Problem 4. Let c n for n N be a non-increasing sequence of positive numbers. Prove that the following two statements are equivalent. a) For any < a < b < the sequence c n sin n n=1 converges uniformly on [a,b] in the sense that given any ε > 0 there eists some N N such that q c n sin n < ε n=p for a b and p, q N). b) lim nc n = 0. n

3 Hint: For a) b), for any n sufficiently large choose p roughly of the order n and choose positive sufficiently close to zero, roughly of the order π, such that n n k=p c k sin k dominates a fied positive number times nc n. For b) a), argue as follows. For π, use summation by parts and p bound c q p n=p sin n by using the summation formula for q n=p sin n. For π, use sin θ < θ for θ > 0 to bound q q n=p c n sin n. For π < < π, q p bound q n=p c n sin n by breaking it up suitably into two summands and use separately the preceding two bounding arguments for the two summands.) Problem 5. Let V be a vector space over F of finite dimension n which is endowed with an inner product,. Let T : V V be an F-linear map which is self-adjoint with respect to, that is, Tv,w = v,tw for all v,w V ). Consider the following procedure. Choose a vector v 1 of unit length in V such that Tv,v achieves its minimum at v = v 1 among all v V of unit length. Inductively suppose v 1,,v k have been chosen and the set v 1,,v k does not span V over F. Let V k be the orthogonal complement of the F-vector subspace of V spanned by v 1,,v k. Choose a vector v k+1 of unit length in V k such that Tv,v achieves its minimum at v = v k+1 among all v V k of unit length. Show that this procedure produces an orthonormal basis v 1,,v n with respect to which T is represented by a diagonal matri with real eigenvalues. Justify carefully each step and eplain why each v k eits. Problem 6. Let V be a vector space over F of finite dimension n. Denote by V the dual vector space of V and regard V as the set of all F-valued F-linear functions on V. Let 1 k n. Define the eterior product k V of k copies of V as the set of all F-valued F-multilinear functions on V V V }{{} k copies which are skew-symmetric in its k variables that is, the value of the function changes sign when any two of the k variables are interchanged). For v 1,,v k V define the wedge product v 1 v k as the F-valued F- multilinear function on V V V }{{} k copies which is the skew-symmetrization of the function 1,,, k ) v 1 1 )v ) v k k ) 3

4 for 1,,, k V. In other words, v 1 v v k ) 1,,, k ) = 1 ) ) ) sign σ)v 1 σ1) v σ) vk σk), k! σ where the summation is over all the k! permutations σ of the k letters {1,,,k} and sign σ) is the signature of the permutation σ. Let, V be an inner product of V. Let e 1,,e n be an orthonormal basis of V over F. Let, k V be the inner product on k V which is defined by the condition that the following collection of n k) elements of k V e j1 e j e jk 1 j 1 < j < < j k n) form an orthonormal basis of k V over F. Show that for u 1,,u k V and v 1,,v k V the inner product u 1 u k, v 1 v k k V of the two elements u 1 u k and v 1 v k of k V is equal to the determinant of the k k matri whose element on the j-th row and in the l-th column is u j, v l V. Hint: Let A be a k n matri and B be an n k matri. For any 1 j 1 < < j k n let A j1,,j k be the k k matri obtained from A by taking only its j-th columns for j = j 1,,j k. Let B j1,,j k be the k k matri obtained from B by taking only its j-th rows for j = j 1,,j k. Epress the k k determinant of AB in terms of the collection of the determinants of A j1,,j k B j1,,j k for all 1 j 1 < < j k n. The special case k = of the problem for u 1 = v 1 = n j=1 a je j and u = v = n j=1 b je j is equivalent to the identity n j=1 ) n ) a j b j n a j b j = j=1 j=1 1 j<l n which is used in the proof of the Cauchy-Schwarz inequality. a k b l a l b k, Problem 7. Let < a < b <. For n N let f n ) be a C-valued continuous function on [a,b] whose first-order derivative f n) is also continuous on [a,b]. Assume that f n a) 1 and a f n) d 1 4

5 for n N. Show that there is a subsequence f nj j N) such that f nj ) f nk ) approach 0 as j, k. Hint: Use y for < y and use gt)ht)dt sup a b y dt f) fy) = ) y ) gt) dt ht) dt y f t)dt to show that for ε > 0 the number δ > 0 chosen in the definition for uniform continuity of f n ) on [a,b] can be chosen to be independent of n.) Problem 8. Let < a < b <. Let X be the set of all C-valued functions f on [a,b] which is continuous on [a,b]. Define the norm f X = sup f) a b for f X. Let Y be the set of all C-valued functions g on [a,b] which is continuous on [a,b] and whose first-order derivative g is also continuous on [a,b]. Define the norm for g Y. g Y = sup g) + g ) ) a b a) Verify that X with the norm X is a Banach space. b) Verify that Y with the norm Y is a Banach space. c) Show that for every sequence g in Y N) with g Y 1 there is a subsequence g j j N) such that as a sequence in X the subsequence g j converges in X to some element of X as j. Hint: for the proof of c) compare with Problem 7.) 5

6 Problem 9. For 0 < < 1 and n N let f n ) be the distance between and the nearest number of the form m, where m Z. Let f) = 10 n n=1 f n). Prove the following two statements. a) The function f) is continuous at every point of 0, 1). b) The function f) is not differentiable at any point of 0, 1). Hint: For the proof of b), for a fied 0, 1) let = q=1 a q 10 q, where a q Z with 0 a q 9. Define q = 1 if a 10 q q = 4 or 9, otherwise define q = + 1. Then 10 q f q ) f) q = q, where q is an integer which is congruent to q 1 modulo.) Problem 10. Suppose < a < b <. Let f) be a bounded real-valued function on [a, b] and α) be a real-valued non-decreasing function [a, b]. Let E be a subset of [a,b]. Assume the following two conditions. a) f is continuous at every point of [a,b] which is not in E. b) For ever ε > 0 there eist a finite number of disjoint open intervals c 1,d 1 ),, c N,d N ) inside [a,b] such that N j=1 αd j) αc j )) < ε and their union N j=1c j,d j ) contains E. Note that in this condition the number N, as well as the intervals c 1,d 1 ),, c N,d N ), may depend on ε.) Prove that f is Riemann-Stieltjes integrable with respect to α on [a,b] that is, in the notation of the book of Rudin, f Rα) on [a,b]). Problem 11. For 1 < s <, define the Riemann zeta function by ζs) = n=1 1 n s. Let [] denote the greatest integer. Prove the following three statements. 6

7 a) b) c) The limit eists for all s > 0. ζs) = [] ζs) = s lim d. b =1 s+1 s s 1 s lim [] d. b =1 s+1 [] lim d b =1 s+1 Hint: To prove a), compute the difference between the integral over [1,N] and the N-th partial sum of the series that defines ζs).) Problem 1. Let < a < b <. Let f) be a real-valued continuous function on [a,b] and φ) be a non-increasing function on [a,b] whose firstorder derivative φ ) is continuous on [a,b]. Show that there eists ξ [a,b] such that ξ f)φ)d = φa) f)d + φb) f)d. =a =ξ =a Hint: First reduce to the special case where φb) = 0. Let F ) = f) with Fa) = 0. Use F) to apply integration by parts to f)φ) and estimate F) by its supremum and infimum on [a,b] and use the Intermediate Value Theorem.) 7