Real Analysis Qualifying Exam May 14th 2016

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1 Real Analysis Qualifying Exam May 4th 26 Solve 8 out of 2 problems. () Prove the Banach contraction principle: Let T be a mapping from a complete metric space X into itself such that d(tx,ty) apple qd(x, y) Written by Prof. S. Lee and Prof. B. Shekhtman for all x, y 2 X and for some q<. For arbitrary x 2 X let x n = Tx n. Prove that x n! x such that Tx = x.(x is a fixed point for T ). Also prove that this is the unique fixed point for T. (2) Does the following it exist dx? ( + x n )n x n If it does, find the it. (3) Let f be a continuous linear functional on a Hilbert space H and M := {x 2 H : f(x) =}. Prove that dim M? apple. (4) Let T be a surjective linear map from a Banach space X onto a Banach space Y such that ktxk 26 kxk for all x 2 X. Show that T is bounded. (5) Let A := span{x n ( x) :n } C([, ]). Describe its closure in the uniform norm. (6) Let µ be a finite positive measure on the measurable space (,,µ) and let F 2 L p (µ) for some apple p<. Prove that there exists g 2 L (µ) such that gdµ = F ( A ) for all A 2. (Here (7) Show that A is the characteristic function of A). A f 2 L (R) () g 2 L (R R) where g(x, y) =f(x + y)f(x). (8) Show that, for apple p<, a bounded sequence {f n } in L p (R) such that {f n }!f pointwise a.e. converges weakly to f in L p (R). (9) Let, µ be finite measures on X. Let F = {f 2 L (X, µ) : fdµ apple (E)} for all measurable sets E. Show that there exists f 2 F such that f dµ =sup fdµ. f2f () Find a bounded sequence in L ([, ]) such that x n! X f n = x E X f for all x 2 [, ] and {f n } does not converge weakly to f in L ([, ]). () Let f be Lipschitz on R and g be absolutely continuous on [, ]. Show that the composition f g is absolutely continuous. (2) Describe the Carathéodory construction of a measure from a set function µ : S! [, ] wheres is a collection of subsets in X. Show that µ is countably monotone if and only if the outer measure induced by µ is an extension of µ.

2 Real Analysis Qualifying Exam September 24th 26 Written by S. Kouchekian and S. Lee INSTRUCTIONS : Do at least 7 problems. Justify your reasoning. State the theorems you use so that all the hypothesis are checked.. Suppose that f is a measurable function in R. Prove that there exists a sequence of step functions that converges pointwise to f(x) for almost every x. 2. Let F : R! R be a function satisfying F (x) = x a f(y)dy for an integrable function. Prove that F is absolutely continuous. 3. Compute the following it and justify the calculation. X /4 n= ( p sin x) n cos xdx. 4. Let (X, M,µ)bea -finite positive measure space and { n } be a sequence of -finite positive measures on M with n µ for all n 2 N such that n (E) apple n+ (E) for all n 2 N, E 2M. Define (E) = n! n (E) for each E 2M. Show that defines a measure on M and satisfies µ. And express d dµ in terms of n. 5. Let f n,f 2 L p [, ] with apple p<. Suppose that f n! f pointwise. Prove that kf n fk p! if and only if kf n k p!kfk p as n!. 6. For each polynomial f on [, ], let kfk = kfk + kf k. Let X be the normed linear space of polynomials on [, ] with the norm k k and Y be the normed linear space of polynomials on [, ] with the norm k k. Let T : X! Y be the linear operator defined by T (f) =f for f 2 X. Show that T is unbounded. The graph of T is the set {(x, T (x)) 2 X Y x 2 X}. Is the graph of T closed in the product topology of X Y? 7. Let (X, M,µ)bea -finite positive measure space and be a finite measure on M. Show that? µ if and only if there is no nonzero measure on M such that µ and apple on M. 8. Given an example of an increasing function on R whose set of discontinuity is precisely Q. 9. Let f 2 L p (R) \ L q (R) withapple p<q<. Prove that f 2 L r (R) for any r with p<r<q.

3 Real Analysis Qualifying Exam January 28th 27 Written by B. Shekhtman and S. Lee INSTRUCTIONS : Do at least 7 problems. Justify your reasoning. State the theorems you use so that all the hypothesis are checked.. Suppose that f and g are continuous function on [, ] and for all n. Prove that f(x) =g(x). f(x)x n dx = g(x)x n dx 2. Let X be a Banach space and Y be a proper closed subspace of X. Prove that for every < " < there exists x 2 X with kxk = such that for all y 2 Y. kx yk > " 3. Let T be a bounded linear operator on a Hilbert space H. Show that kt k = kt k and kt T k = kt k Let f be a non-negative function in L ([, ]). Show that n! (Here m stands for Lebegues measure on [, ]). p n f(x)dx = m{x 2 [, ] : f(x) > }. 5. Let f be a continuous function on [, ]. Show that n! R (n + )xn f(x)dx = f(). 6. Let (X, B,µ) be a measure space and f 2 L p (µ) \ L (µ). Show that p! kfk p = kfk 7. Suppose {f n } n= is a sequence of nonnegative measurable functions on [, ] with f n dx apple n 2 for all n.

4 Prove that f n! a.e. on [, ]. 8. Let f 2 L (R). Show that g(x) = R cos(xt)f(t)dt is continuous and bounded. 9. Let {E n } be a sequence of Lebesgue measurable subsets of R with a) Show that X µ(e n ) <. () n= where sup n! E n = T m= S n=m E n µ( sup E n )=, n! b) Is the conclusion in a) still true if () is replaced by P n= (µ(e n)) 2 <?. Let <a<b. Show that the function f : R 2! R given by ( e xy (x, y) 2 [, ) [a, b], f(x, y) = otherwise is integrable with respect to the Lebesgue measure on R 2. Compute the integral e ax x e bx dx.. Prove f : R! R is absolutely continuous over any compact set if and only f g is absoletely continuous for all smooth g : R! R supported on a compact set. 2. Let E [, ] be measurable. Define a function f :[, ]! R by f(x) = x ( 2 E (t))dt. Prove that f is of bounded variation and determine the total variation of f on [, ]. 2

5 QUALIFYING EXAM: REAL ANALYSIS May 3, 27 Prof. C. Bénéteau Prof. S.-Y. Lee Answer 4 questions from Part A and 3 questions from Part B. Part A.. Let m be Lebesgue measure on R, andsupposethate [, ] is a set of real numbers with the property that for any x and y in E with x 6= y, x y is not equal to a rational number. Prove that the collection {E + r : r 2 Q \ [4, 5]} is a countable collection of mutually disjoint sets. Prove that either m(e) =ore is not measurable. 2. Let X =[, ] and let m be Lebesgue measure on X. Definewhatitmeans for a sequence of measurable functions f n to converge to a function f in measure. Prove or disprove the following statement: If f n converges to f in measure, then f n converges to f pointwise a.e. 3. Consider the function f(x) =x cos(/ p x)definedontheinterval[, ], where we set f() =. Is f of bounded variation? Prove or disprove. 4. (a) Given a measure space (X, M,µ)andasequenceofintegrablefunctions {f n } n=, define what it means for the sequence to be uniformly integrable. (b) If X =[, ), M is all Lebesgue measurable sets, and µ is Lebesgue measure, is the sequence f n (x) = p n [,/n) (x)uniformlyintegrable? Prove or disprove. 5. (a) State the Radon-Nikodym Theorem. (b) Give an example that shows the necessity of the requirement that the space X be finite in the Radon-Nikodym theorem. (c) Suppose X =[, ], M is the -algebra of Lebesgue measurable sets, µ is Lebesgue measure on X, and is counting measure on X. Prove or disprove each of the following statements: () µ<< ;(2) <<µ;(3)µ and are mutually singular. 6. Suppose {f n } n= is a sequence of nonnegative measurable functions on [, ] with f n dx apple a n for all n, where P n= a n <. Prove that f n! a.e.on[, ].

6 7. Let F : R! R be a function satisfying F (x) = x a f(y)dy for an integrable function f. Prove (without referring to any theorems) that F is absolutely continuous. Part B. 8. Let f n ( ) =e in for apple apple 2. Show that f n ( )! weaklyinl p [, 2 ] for any <p<. 9. Let µ be a complex measure µ on a measure space (X, M), i.e., µ is a countably additive set function from M to C satisfying µ(;) =. Define a new set function µ on M as follows: ( ) X µ (E) := sup µ(e k ) : E k 2M are pairwise disjoint,e=[ k=e k. k= Prove that µ is a positive measure. variation measure.) ( µ is sometimes called the total. Let (X, M,µ)beafinitemeasurespace. ShowthatL p (X, µ) ( L (X, µ) for any p>. Is it true if we remove the hypothesis that µ(x) <?. Let X = Y =[, ] and for each positive integer n, let' n be the characteristic function of the interval [ /2 n, /2 n ), and let g n =2 n ' n. Define X f(x, y) = [g n (x) g n+ (x)] g n (y). n= Prove that f is measurable as a function on the product space X Y (with respect to Lebesgue measure), and calculate each of the iterated integrals R R f(x, y) dy dx and R R f(x, y) dx dy. Is the function f(x, y) X Y Y X integrable on [, ] [, ]? Justify fully. 2. Let f 2 L (R). Show that g(x) = R sin(xt)f(t)dt is continuous and bounded. 3. Let T be a surjective linear map from a Banach space X onto a Banach space Y such that ktxk 27 kxk for all x 2 X. ShowthatT is bounded. 4. Find a linear functional L on C[ 2, 2] such that there does not exist a function g 2 C[ 2, 2] with kgk =and L(g) = klk.

7 Prof. C. Bénéteau Prof. S.-Y. Lee Prof. B. Shekhtman QUALIFYING EXAM: REAL ANALYSIS September 3, 27 Answer 4 questions from Part A and 3 questions from Part B. Part A.. Let X be a set and B a -algebra of subsets on X. Let {µ n } n= be a sequence of (positive) measures on (X, B) suchthatµ n+ (E) µ n (E) for every E 2B.Letµ(E) := n! µ n (E). Prove that µ is a measure on B. Is this conclusion still true if we assume that µ n+ (E) apple µ n (E)? 2. Consider the function f(x) = x 2 sin(/x) definedontheinterval[, ], where we set f() =. Is f of bounded variation? Prove or disprove. 3. State and prove Fatou s lemma, and give an example where strict inequality occurs. 4. (a) Given a measure space (X, M,µ)andasequenceofintegrablefunctions {f n } n=, define what it means for the sequence to be uniformly integrable. (b) If X =[, ), M is all Lebesgue measurable sets, and µ is Lebesgue measure, is the sequence f n (x) =n 2 [,/n 3 )(x)uniformlyintegrable? Prove or disprove. 5. State the Hahn Decomposition Theorem. Consider the set X =[ /2, /2], and let M be the -algebra of Lebesgue measurable sets on X. Let f(t) = sin(2t). Define, for any measurable set E, µ(e) = f(t) dt, where dt is the usual Lebesgue measure. Find the Hahn Decomposition for this measure, and compute µ + (X), µ (X), and the total variation µ (X). 6. Let f and g be continuous functions on [, ] such that f(x) <g(x) forall x 2 [, ]. Prove that there exists a sequence of polynomials p n such that E f(x) <p n (x) <g(x) for all n, andp n! g uniformly on [, ]. 7. Let A =span{x n ( x),n =, 2,...} C([, ]). Determine the closure of A in the uniform norm.

8 Part B. 8. Define what a complex measure µ on a measure space (X, M) is,and define what the total variation µ is. Prove that the total variation µ is a (positive) measure. 9. Let (X, M,µ)beameasurespace.Forapple p<, define what L p (X) is, and state and prove Hölder s inequality.. Show that Justify fully. [,] [,] dxdy <. xy. Let H be a Hilbert space, and let T be a continuous linear operator on H. Let T be the operator defined by ht x, yi = hx, T yi for all x, y 2 H. Prove that T is invertible if and only if T is. 2. Let X be a Banach space, and define the mapping J : X! X by J(x)(f) =f(x) for each f 2 X and each x 2 X. The space X is called reflexive if J is surjective. Prove that on a reflexive Banach space, every functional attains its norm, that is, for every f 2 X, there exists x 2 X with kxk =such that f(x) =kfk. 3. Let f : R! R be a smooth function with a compact support. Show that, for each positive integer N, sin(tx)f(t)dt apple Cx N, x > for some constant C>. R

9 Real Analysis Qualifying Exam January 27th 28 Solve 8 out of 2 problems. Written by S.-Y. Lee and B. Shekhtman. Assume that the function f is of bounded variation on [, ]. For each x 2 [, ], define v(x) tobethe total variation of the restriction of f to [,x]. Prove that if f is absolutely continuous then v is absolutely continuous. 2. Let {f n } be a sequence of nonnegative Lebesgue measurable functions on [, ]. Show that {f n } converges to zero in measure if and only if n! f n (x) dx =. +f n (x) 3. Assume that the functions f and g are both integrable and bounded on R. Define the function h on R by h(x) = f(x + y)g(y)dy. Prove that h is a continuous function on R, and that x! h(x) =. 4. Let f and g be real valued measurable functions on [, ] with the property that for every x 2 [, ], g is di erentiable at x and g (x) =f(x) 2. Prove that f 2 L [, ]. 5. Let f : R! R be a C function with a compact support. Show that, for each positive integer N, sin(tx)f(t)dt apple Cx N, x > for some constant C>. R 6. Show that Justify your answer. [,] [,] dxdy <. xy

10 7. Let f n be a sequence of functions in L ([, ]) with respect to Lebegues measure such that, as n tends to infinity, f n (x) dx!. Does it follow that f n! a.e.. Give a proof or a counterexample. 8. Describe all intervals [a, b] R such that span{x 2n,n=,, 2,...} is dence in C[a, b]. 9. Show that for every continuous function f on [, ) f(x)e nx dx =. n!. Let T be a linear map from a Banach space X onto a Banach space Y such that ktxk kxk for all x 2 X. Show that T is a bounded linear operator.. Let Y be a closed subspace of C([, ) and g/2 Y. Prove that there exists a signed Borel measure µ on [, ] such that fdµ = for all f 2 Y and R gdµ =. 2. Show that l is not a separable space. 2