Real Analysis Qualifying Exam – May 14Th 2016 Written by Prof

Home , Uniform norm

Real Analysis Qualifying Exam – May 14th 2016 Written by Prof. S. Lee and Prof. B. Shekhtman Solve 8 out of 12 problems. (1) Prove the Banach contraction principle: Let T be a mapping from a complete metric space X into itself such that d(Tx,Ty) qd(x, y)  for all x, y X and for some q<1. For arbitrary x0 X let xn = Txn 1. Prove that xn x such that 2 2 ! Tx = x.(x is a fixed point for T ). Also prove that this is the unique fixed point for T . (2) Does the following limit exist 1 dx lim ? x 1 (1 + )nx n Z0 n If it does, find the limit. (3) Let f be a continuous linear functional on a Hilbert space H and M := x H : f(x)=0 . Prove that { 2 } dim M ? 1.  (4) Let T be a surjective linear map from a Banach space X onto a Banach space Y such that 1 Tx x k k2016 k k for all x X. Show that T is bounded. 2 (5) Let A := span xn(1 x):n 1 C([0, 1]). Describe its closure in the uniform norm. { }⇢ (6) Let µ be a finite positive measure on the measurable space (⌦, ⌅,µ) and let F L (µ)⇤ for some 2 p 1 p< . Prove that there exists g L (µ) such that  1 2 1

gdµ = F (A) ZA for all A ⌅. (Here is the characteristic function of A). 2 A (7) Show that 1 1 f L (R) g L (R R) 2 () 2 ⇥ where g(x, y)=f(x + y)f(x). p (8) Show that, for 1 p< , a bounded sequence fn in L (R) such that fn f pointwise a.e.  1 { } { }! converges weakly to f in Lp(R). (9) Let , µ be ﬁnite measures on X. Let

F = f L1(X, µ): fdµ (E) { 2  } ZE for all measurable sets E. Show that there exists f F such that 0 2

f0dµ =sup fdµ. X f F X Z 2 Z (10) Find a bounded sequence in L1([0, 1]) such that x x lim fn = f for all x [0, 1] n 2 !1 Z0 Z0 and f does not converge weakly to f in L1([0, 1]). { n} (11) Let f be Lipschitz on R and g be absolutely continuous on [0, 1]. Show that the composition f g is absolutely continuous. (12) Describe the Carath´eodory construction of a measure from a set function µ : S [0, ]whereS is a ! 1 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.

1 Real Analysis Qualifying Exam – September 24th 2016

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.

1. 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 ! x F (x)= f(y)dy Za for an integrable function. Prove that F is absolutely continuous.

3. Compute the following limit and justify the calculation.

⇡/4 1 (1 psin x)n cos xdx. n=0 0 X Z 4. Let (X, ,µ)bea-ﬁnite positive measure space and ⌫ be a sequence of -ﬁnite positive M { n} measures on with ⌫n µ for all n N such that M ⌧ 2

⌫n(E) ⌫n+1(E) for all n N,E .  2 2M

Define ⌫(E)=limn ⌫n(E) for each E . Show that ⌫ defines a measure on and ⌫ satisfies !1 2M M ⌫ µ. And express d⌫ in terms of ⌫ . ⌧ dµ n 5. Let f ,f Lp[0, 1] with 1 p< . Suppose that f f pointwise. Prove that f f 0if n 2  1 n ! k n kp ! and only if f f as n . k nkp !k kp !1 6. For each polynomial f on [0, 1], let f = f + f 0 . k k k k1 k k1 Let X be the normed linear space of polynomials on [0, 1] with the norm and Y be the normed k·k1 linear space of polynomials on [0, 1] with the norm . Let T : X Y be the linear operator k·k ! defined by T (f)=f 0 for f X. Show that T is unbounded. 2 The graph of T is the set (x, T (x)) X Y x X . Is the graph of T closed in the product { 2 ⇥ | 2 } topology of X Y ? ⇥ 7. Let (X, ,µ)bea-finite positive measure space and ⌫ be a finite measure on . Show that ⌫ µ M M ? if and only if there is no nonzero measure ⇢ on such that ⇢ µ and ⇢ ⌫ on . M ⌧  M 8. Given an example of an increasing function on R whose set of discontinuity is precisely Q.

9. Let f Lp(R) Lq(R)with1 p

1 Real Analysis Qualifying Exam – January 28th 2017 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.

1. Suppose that f and g are continuous function on [0, 1] and

1 1 f(x)xndx = g(x)xndx Z0 Z0 for all n 0. Prove that f(x)=g(x).

2. Let X be a Banach space and Y be a proper closed subspace of X. Prove that for every 0 < " < 1 there exists x X with x = 1 such that 2 k k x y > " k k for all y Y . 2

2 3. Let T be a bounded linear operator on a Hilbert space H. Show that T = T ⇤ and T ⇤T = T . k k k k k k k k

4. Let f be a non-negative function in L1([0, 1]). Show that 1 lim n f(x)dx = m x [0, 1] : f(x) > 0 . n !1 0 { 2 } Z p (Here m stands for Lebegues measure on [0, 1]).

1 n 5. Let f be a continuous function on [0, 1]. Show that limn (n + 1)x f(x)dx = f(1). !1 0 R

p 6. Let (X, ,µ) be a measure space and f L (µ) L1(µ). Show that B 2 \ lim f = f p p !1 k k k k1

7. Suppose f 1 is a sequence of nonnegative measurable functions on [0, 1] with { n}n=1 1 1 f dx for all n 1. n  n2 Z0

1 Prove that f 0 a.e. on [0, 1]. n !

8. Let f L1(R). Show that g(x)= 1 cos(xt)f(t)dt is continuous and bounded. 2 1 R

9. Let En be a sequence of Lebesgue measurable subsets of R with { }

1 µ(E ) < . (1) n 1 n=1 X a) Show that µ(lim sup En)=0, n !1

where lim supn En = m1=1 n1=m En !1 2 b) Is the conclusion in a) still true if (1) is replaced by 1 (µ(E )) < ? T S n=1 n 1 P

10. Let 0

is integrable with respect to the Lebesgue measure on R2. Compute the integral

ax bx 1 e e dx. x Z0

11. 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. !

12. Let E [0, 1] be measurable. Deﬁne a function f :[0, 1] R by ⇢ ! x f(x)= (1 2 (t))dt. E Z0 Prove that f is of bounded variation and determine the total variation of f on [0, 1].

2 QUALIFYING EXAM: REAL ANALYSIS May 13, 2017

Prof. C. B´en´eteau Prof. S.-Y. Lee

Answer 4 questions from Part A and 3 questions from Part B. Part A.

1. Let m be Lebesgue measure on R,andsupposethatE [0, 1] is a set of real numbers with the property that for any x and y in E⇢with x = y, x y is not equal to a rational number. Prove that the collection E6 + r : r { 2 Q [4, 5] is a countable collection of mutually disjoint sets. Prove that either\ m(}E)=0orE is not measurable.

2. Let X =[0, 1] and let m be Lebesgue measure on X.Deﬁnewhatitmeans for a sequence of measurable functions fn to converge to a function f in measure. Prove or disprove the following statement: If fn converges to f in measure, then fn converges to f pointwise a.e. 3. Consider the function f(x)=x cos(1/px)deﬁnedontheinterval[0, 1], where we set f(0) = 0. Is f of bounded variation? Prove or disprove.

4. (a) Given a measure space (X, ,µ)andasequenceofintegrablefunctions M f 1 , deﬁne what it means for the sequence to be uniformly integrable. { n}n=1 (b) If X =[0, ), is all Lebesgue measurable sets, and µ is Lebesgue 1 M measure, is the sequence fn(x)=pn[0,1/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 =[0, 1], is the -algebra of Lebesgue measurable sets, µ is Lebesgue measure onMX,and⌫ is counting measure on X. Prove or disprove each of the following statements: (1) µ<<⌫;(2)⌫<<µ;(3)µ and ⌫ are mutually singular.

6. Suppose fn n1=1 is a sequence of nonnegative measurable functions on [0, 1] with { } 1 f dx a for all n 1, n  n Z0 where 1 a < . Prove that f 0a.e.on[0, 1]. n=1 n 1 n ! P 7. Let F : R R be a function satisfying ! x F (x)= f(y)dy Za for an integrable function f.Prove(withoutreferringtoanytheorems) that F is absolutely continuous. Part B.

in✓ p 8. Let fn(✓)=e for 0 ✓ 2⇡. Show that fn(✓) 0weaklyinL [0, 2⇡] for any 1

10. Let (X, ,µ)beaﬁnitemeasurespace.ShowthatLp(X, µ) ( L1(X, µ) for any p>M 1. Is it true if we remove the hypothesis that µ(X) < ? 1 11. Let X = Y =[0, 1] and for each positive integer n,let'n be the charac- n 1 n n teristic function of the interval [1 1/2 , 1 1/2 ), and let gn =2 'n. Deﬁne 1 f(x, y)= [g (x) g (x)] g (y). n n+1 n n=1 X 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 X Y f(x, y) dy dx and Y X f(x, y) dx dy. Is the function f(x, y) integrable on [0, 1] [0, 1]? Justify fully. R R ⇥ R R 12. Let f L1(R). Show that g(x)= 1 sin(xt)f(t)dt is continuous and bounded.2 1 R 13. Let T be a surjective linear map from a Banach space X onto a Banach space Y such that 1 Tx x k k2017 k k for all x X.ShowthatT is bounded. 2 14. Find a linear functional L on C[ 2, 2] such that there does not exist a function g C[ 2, 2] with g =1and L(g) = L . 2 k k1 | | k k QUALIFYING EXAM: REAL ANALYSIS September 30, 2017

Prof. C. B´en´eteau Prof. S.-Y. Lee Prof. B. Shekhtman

Answer 4 questions from Part A and 3 questions from Part B. Part A.

1. Let X be a set and a -algebra of subsets on X.Let µn n1=1 be a sequence of (positive)B measures on (X, )suchthatµ (E){ }µ (E)for B n+1 n every E .Letµ(E) := limn µn(E). Prove that µ is a measure on . !1 Is this conclusion2B still true if we assume that µ (E) µ (E)? B n+1  n 2. Consider the function f(x)=x2 sin(1/x)deﬁnedontheinterval[0, 1], where we set f(0) = 0. 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, ,µ)andasequenceofintegrablefunctions M f 1 , deﬁne what it means for the sequence to be uniformly integrable. { n}n=1 (b) If X =[0, ), is all Lebesgue measurable sets, and µ is Lebesgue 1 M 2 measure, is the sequence fn(x)=n [0,1/n3)(x)uniformlyintegrable?Prove or disprove.

5. State the Hahn Decomposition Theorem. Consider the set X =[ ⇡/2,⇡/2], and let be the -algebra of Lebesgue measurable sets on X.Let f(t)= sin(2t). MDeﬁne, for any measurable set E,

µ(E)= f(t) dt, ZE 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 [0, 1] such that f(x)

f(x)

for all n,andp g uniformly on [0, 1]. n ! 7. Let A =span xn(1 x),n =1, 2,... C([0, 1]). Determine the closure of A in the uniform{ norm. }⇢ Part B.

8. Deﬁne what a complex measure µ on a measure space (X, )is,and deﬁne what the total variation µ is. Prove that the total variationM µ is a (positive) measure. | | | |

9. Let (X, ,µ)beameasurespace.For1 p< , deﬁne what Lp(X)is, and stateM and prove H¨older’s inequality.  1

10. Show that 1 dxdy < . [0,1] [0,1] 1 xy 1 ZZ ⇥ Justify fully.

11. Let H be a Hilbert space, and let T be a continuous linear operator on H. Let T ⇤ be the operator deﬁned by

T ⇤x, y = x, T y for all x, y H. h i h i 2

Prove that T is invertible if and only if T ⇤ is.

12. Let X be a Banach space, and deﬁne the mapping J : X X⇤⇤ by ! J(x)(f)=f(x)

for each f X⇤ and each x X. The space X is called reﬂexive if J is surjective. Prove2 that on a reﬂexive2 Banach space, every functional attains its norm, that is, for every f X⇤, there exists x X with x =1such that f(x)= f . 2 2 k k k k 13. Let f : R R be a smooth function with a compact support. Show that, for each positive! integer N,

N sin(tx)f(t)dt Cx ,x>0  ZR for some constant C>0. Real Analysis Qualifying Exam – January 27th 2018 Written by S.-Y. Lee and B. Shekhtman Solve 8 out of 12 problems. 1. Assume that the function f is of bounded variation on [0, 1]. For each x [0, 1], deﬁne v(x)tobethe 2 total variation of the restriction of f to [0,x]. Prove that if f is absolutely continuous then v is absolutely continuous.

2. Let f be a sequence of nonnegative Lebesgue measurable functions on [0, 1]. Show that f { n} { n} converges to zero in measure if and only if 1 f (x) lim n dx =0. n 1+f (x) !1 Z0 n

3. Assume that the functions f and g are both integrable and bounded on R. Deﬁne the function h on R by 1 h(x)= f(x + y)g(y)dy. Z1 Prove that h is a continuous function on R, and that lim x 0 h(x) = 0. | |!

4. Let f and g be real valued measurable functions on [0, 1] with the property that for every x [0, 1], g 2 is di↵erentiable at x and 2 g0(x)=f(x) . Prove that f L1[0, 1]. 2

5. Let f : R R be a C1 function with a compact support. Show that, for each positive integer N, ! N sin(tx)f(t)dt Cx ,x>0  ZR for some constant C>0.

6. Show that 1 dxdy < . [0,1] [0,1] 1 xy 1 ZZ ⇥ Justify your answer.

1 7. Let fn be a sequence of functions in L1([0, 1]) with respect to Lebegues measure such that, as n tends to inﬁnity, f (x) dx 0. | n | ! Z Does it follow that f 0 a.e.. Give a proof or a counterexample. n !

8. Describe all intervals [a, b] R such that span x2n,n=0, 1, 2,... is dence in C[a, b]. ⇢ { }

9. Show that for every continuous function f on [0, ) 1

nx lim f(x)e dx = 0. n 0 ! Z

10. Let T be a linear map from a Banach space X onto a Banach space Y such that 1 Tx x k k10 k k for all x X. Show that T is a bounded linear operator. 2

11. Let Y be a closed subspace of C([0, 1) and g/Y . Prove that there exists a signed Borel measure µ 2 on [0, 1] such that fdµ =0 Z for all f Y and gdµ = 1. 2 R

12. Show that l is not a separable space. 1