Mathematics Department, Stanford University Real Analysis Qualifying Exam, Autumn 1998 Part I

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1 Mathematics Department, Stanford University Real Analysis Qualifying Exam, Autumn 1998 Part I Do all five problems (Use a different blue book for each problem) 1. Suppose f : R R is differentiable at, and suppose a n and b n are sequences converging to with a n < b n for all n. (a) If a n < < b n for all n, prove that f(b n ) f(a n ) b n a n f (). (b) Prove that if f is differentiable in a neighborhood of and if f is continuous at, then (*) holds for all sequences a n and b n converging to with b n > a n. (c) Give an example of a function f such that (*) holds for all sequences a n < b n tending to, but for which there are points arbitrarily close to at which f is not differentiable. 2. Suppose f : [, 1] R is a bounded Lebesgue measurable function. Suppose for every x [, 1] there is a function g x such that f = g x a.e. and such that lim g x(t) exists. t x Prove that there is a continuous function g such that g = f almost everywhere. 3. Let f : R R be an L 1 function. Show that f and its Fourier transform cannot both have compact support (unless f = a.e.). 4. Let X be an infinite-dimensional Banach space. (a) Let S be a subset of X such that the linear span of S X (that is, the set of all linear combinations of finite subsets of X) is all of X. Prove that S is uncountable. (b) Suppose the dual space X of X is separable. Prove that X is separable. (c) Let P be a finite-dimensional subspace of X. Prove that there is a bounded linear projection π : X P (in other words, prove that there is a bounded linear operator π : X P such that π(x) = x if x P.) 5. Let f n : [, 1] R be a sequence of continuous functions converging pointwise to a continuous function g. Suppose f n (x) f n+1 (x) for every n and every x [, 1]. Prove that f n f uniformly. 1

2 Mathematics Department, Stanford University Real Analysis Qualifying Exam, Autumn 1998 Part II Do all five problems (Use a different blue book for each problem) 1. Let f : R R be a convex function. That is, suppose f(tx + (1 t)y) tf(x) + (1 t)f(y) for all t (, 1) and for all x and y. (a) Prove that f is continuous everywhere. (b) Prove that f is differentiable except at a countable set of points. (c) Suppose f is strictly convex. (That is, suppose the inequality (*) is strict whenever x y and < t < 1.) If u : [, 1] R is an L 1 function, Jensen s theorem says ( f ) u f(u). Prove that if we have equality, then u is equal a.e. to a constant function. 2. Suppose f : (, ) R is a continuous function such that lim n f(n2 x) = a for every x. (Of course here n is an integer.) Prove that lim x f(x) = a. 3. Suppose f : [, 1] R is a Lebesgue measurable function. (a) Show that the image {f(x) : x [, 1]} need not be a Lebesgue measurable set. (b) Show that there is a function g which is equal to f almost everywhere and such that the image under g of any closed subset of [, 1] is an F σ set (i.e., a countable union of closed sets). 4. (a) If f and g are in L 2 (T), prove that f g is continuous. (b) Construct a continuous function g on T such that g g g (k times) is not in C 1 (T) for any k. 5. Let < a < b < and suppose B is a countable collection of closed subintervals of (a, b). Give the proof that there is a countable pairwise-disjoint subcollection B B such that I B Ĩ I B I. Here Ĩ denotes the 5-times enlargment of I; thus if I = [x ρ, x+ρ] then Ĩ = [x 5ρ, x + 5ρ]. 2

3 Real Analysis Exam: Part I (Spring 1998) Do all five problems. 1. Suppose f n, f L 2 ([, 1]) with f n f in measure and with f n L 2 1 for all n 1. (a) Prove (i) that f n f weakly in L 2, and (ii) that f n f L 2 if and only if f n L 2 f L 2. (b) Prove that f n f L p for each p [1, 2). 2. Let f : R R be nowhere continuous. Prove that there is an ɛ > and a nonempty open interval (a, b) such that lim sup t x f(t) lim inf t x f(t) ɛ for every x (a, b). 3. Consider the function f(x) = n=1 1 n cos(1 2n x). (a) Prove that there is a C > such that f(x) f(y) C x y 1/2 for every x, y R. (b) Prove that f is nowhere differentiable. Hint: consider the choices h = 1 2m π and h = 1 2m π 2 in f(x+h) f(x) h, m = 1, 2, If S is an uncountable subset of C([, 1]), prove that there is a uniformly convergent sequence {f n } n=1,2,... of distinct functions in S. 5. Suppose µ is an outer measure on a separable space X such that all Borel sets are µ-measurable and such that µ(x) <. Let S = {x X : lim sup ρ 1 µ(b ρ (x)) = }. ρ Prove that S has 1-dimensional measure. That is, prove that for each ɛ >, there is a covering of S by balls B ρj (x j ) such that j ρ j < ɛ. Note: you may assume without proof the five times covering lemma, which says that if B is a collection of closed balls in X such that sup{diam B : B B} <, then there is a countable pairwise disjoint subcollection {B ρj (x j )} B such that B B B j B 5ρj (x j ).

4 Real Analysis Exam: Part II (Spring 1998) Do all five problems. 1. Two quickies: (i) Let E be a compact subset of C([, 1]), where C([, 1]) is equipped with the usual sup norm. Prove that E is equicontinuous. (ii) Let f = f 1 + if 2 be a complex-valued L 1 (R) function. Prove that f(x) dx R R f(x) dx. Also, if R f(x) dx = 1 and if f(ξ) = R e ixξ f(x) dx, prove the strict inequality f(ξ) < 1 for all but possibly one value of ξ R. 2. For t >, let F (t) = χ [,t] (the indicator function of the interval [, t]). (a) Prove that F, as a map from (, ) to L 2 (R), is nowhere differentiable. (That is, F (t+h) F (t) lim h h never exists as a limit taken in L 2 (R) for t (, ).) (b) If g : L 2 (R) R is a bounded linear functional, prove that g F is differentiable almost everywhere on (, ). 3. In this problem, let L 2 ([, 1]) denote the complex Hilbert space of square integrable complex-valued functions f on [, 1] with the usual inner product f, g = 1 f(t)g(t) dt. Define T : L 2 ([, 1]) L 2 ([, 1]) by T f(x) = x f(t) dt for x [, 1]. (a) Prove that T is a compact continuous map. (b) Prove that T has no eigenvalues. That is, prove there is no λ C such that T (f) = λf for some nonzero f L 2 ([, 1]). (c) Prove that the spectrum of T is {}. That is, f T (f) λf is an isomorphism of L 2 ([, 1]) onto L 2 ([, 1]) for each nonzero λ, and it is not such an isomorphism for λ =. 4. Suppose f : R R is 2π-periodic and f L 2 ( π, π). Supose also that ( f(n) n k ) 2 < n Z for some integer k 1. Prove that f is almost everywhere equal to a C(R) function if k = 1 and to a C k 1 (R) function if k 2. 5.(a) Let {x n } n=1,2,... be a sequence in a Banach space B, and let X be the convex hull. In other words, N N X = λ j x j : N 1, λ j [, 1], λ j = 1. j=1 If x n converges weakly to some x in B, prove that some sequence {y n } X converges to x strongly (i.e., y n x ). (b) Give an example of a Hilbert space H and a sequence x n weakly converging to zero in H such that n 1 n j=1 x j does not converge to zero as n. j=1

5 Real Analysis Exam: Part I (Fall 1999) Do all five problems. 1. Suppose f n : [, 1] R is a sequence of absolutely continuous functions such that f n = and such that f n p 1 for some p > 1. Prove that there is a uniformly converging subsequence. 2. Let K be a compact subset of R n and C be a vector space of continuous functions on R n such that for each u C, u() max x K u(x). Prove that there is a measure µ on K such that for every u C. u() = 3. Suppose f and g are positive, measurable functions on [, 1] such that f(x)g(x) 1 for every x. Prove that f(x) dx g(x) dx 1. K u dµ 4. Show that in the Banach space C[, 1], the functions of class C 1 form a set of the first category. 5. Suppose f : N R is a function such that f(n + m) f(n) + f(m) for all n, m N. Prove that exists and is equal to f(n) lim n n f(n) inf n N n.

6 Real Analysis Exam: Part II (Fall 1999) Do all five problems. 1. Prove or disprove: (a) There is a bounded linear functional T : L (R) R such that T (u) = u() provided u is continuous at. (b) There is a bounded linear functional S : L (R) R such that provided u is differentiable at. S(u) = u () 2. Let S be an uncountable subset of L 2 (R). Prove that there exists a sequence u i in S that converges (in L 2 ) to a limit u. 3. Suppose g is an L 1 function on [, 1] such that 1 for every smooth function f : [, 1] R. ( 1 ) 1/2 f g f 2 Prove g is equal almost everywhere to a function that is differentiable almost everywhere. 4. Suppose F : R k R n is a continuous, surjective map. Prove that there is an r > such that the image of B(, r) contains a nonempty open subset of R n. 5. Let X be a compact metric space and µ be a finite, nonnegative Borel measure on X. Suppose µ{x} = for every x X. Prove that for every ɛ >, there is a δ > such that µ(e) < ɛ whenever E is a Borel set in X having diameter δ.

7 Do all five problems. Real Analysis Exam: Part I (Spring 1999) 1. Write the Fourier series of the function f(x) = x 2 on the interval [ π, π], and use it to compute n=1 n Let B be a Banach space and L : B B be a linear (but not necessarily bounded) operator such that Lx x for all x B. Let f : B R be a bounded linear functional. Prove that there is a bounded linear functional g : B R such that g(l(x)) = f(x) for all x B. 3. Let T = R/2πZ be the circle group. Let k L 1 (T) and let K be the integral operator on L 2 (T) defined by K : f 1 k(x t)f(t) dt 2π t T (a) Prove that K is a compact operator (i.e., that the image of the unit ball under K has compact closure.) (b) Find the eigenvalues and the corresponding eigenfunctions of K. Hint: use Fourier series. 4. If f L 1 (R), let M f denote the Hardy-Littlewood maximal function of f: Prove that 1 x+h M f (x) = sup f(t) dt h> 2h x h µ{x : M f (x) > λ} C f 1 λ where f 1 is the L 1 norm of f and C is a fixed constant. 5. Let F : [, 1] R be a continuous function. The arclength L of the graph of F is defined to be n sup xi 2 + y i 2 i=1 where x i = x i x i 1 and y i = F (x i ) F (x i 1 ), and where the supremum is taken over all partitions = x x 1 x n = 1. In this problem, F is increasing, F () =, and F (1) = 1. (a) Prove that 2 L 2. (b) Let µ be the measure such that µ((a, b]) = F (b) F (a). Suppose µ is singular with respect to Lebesgue measure. Prove that L = 2.

8 Do all five problems. Real Analysis Exam: Part II (Spring 1999) 1. Let S be the set of real numbers x [, 1] such that for every ɛ >, there is a rational number p/q with x p q < ɛ q 3 (a) Prove that S is uncountable. (b) Prove that S has measure. 2. Let X be a compact Hausdorff space and J be an ideal in C(X), the space of continuous real-valued functions on X. Assume that for each point x X, there is a function f J such that f(x). Prove that J = C(X). 3. Suppose f : [, 1] R is a function such that lim sup h f(x + h) f(x) h < for almost every x. Prove that for each ɛ >, there is a Lipschitz function g : [, 1] R such that {x : f(x) g(x)} has Lebesgue measure < ɛ. NOTE: You may use (without proof) the following fact. If S R and h : S R is Lipschitz, then there is a Lipschitz function H : R R such that H(x) = h(x) for all x S. 4. Let X and Y be Hilbert spaces and L : X Y be a linear operator. Prove that the following two conditions are equivalent: (a) The image L(B) of the unit ball in X has compact closure in Y, (b) There is a sequence of bounded linear maps L n : X Y such that the image L n (X) is finite dimensional and such that L n L. (Here is the operator norm.) 5. Let f be a continuous real-valued function on [, 1] and let g(y) be the number of points x such that f(x) = y. Prove that if g is in L 1, then f has bounded variation.

9 Do all five problems. Real Analysis Exam: Part I (Autumn 2) 1. Let A, B R be Lebesgue measurable subsets with finite positive measure. (a) Show that the convolution χ A χ B of their characteristic functions is a continuous function that is not identically zero. (b) Show that the set A + B = {a + b : a A, b B} contains a nonempty open interval. 2. Suppose {f n } n=1 is a norm-bounded sequence of functions in L 2 (, 1) that converges in measure to a function f. 3. (a) Show that f L 2 (, 1) and f 2 lim inf f n 2. (b) Show that f n 2 converges to f 2 if and only if f n f 2. (a) Let g(t) be a continuous function in L 2 (R). Assume that g(t) for almost every t R. Show that the linear span of the functions {g(t)e itx : x R} is dense in L 2 (R). (Hint: Hahn-Banach and Fourier Inversion.) (b) Suppose f L 2 (R) and let f denote its L 2 Fourier transform. Assume that f(t) for almost every t R. Show that the linear span of the set of functions {f y (x) = f(x + y) : y R} is dense in L 2 (R). 4. Show that there is no sequence of positive continuous functions {f n } such that {f n (x)} is bounded for each irrational x, and unbounded for each rational x. 5. (a) Let A be a closed bounded subset of a Hilbert space H. Show there is a unique smallest ball in H containing the set A. The center of this ball is called the circumcenter of A. Show that the circumcenter of A lies in A if A is a convex set. (b) Let {A i } be a nested decreasing sequence of nonempty closed bounded convex sets in the Hilbert space H. Show that the sequence {p i } of circumcenters of the sets A i is a Cauchy sequence in H. Show that the limit of this sequence lies in the intersection of the A i. (c) Construct an example of a Banach space X containing a nested decreasing sequence of closed bounded convex sets A i such that i=1 A i = φ. (Hint: You could try X = c, the space of sequences of real numbers converging to zero, with the maximum norm.)

10 Real Analysis Exam: Part II (Autumn 2) Do all five problems. 1. Let (X, ρ) be a metric space, and let C 1 (X) denote the space of continuous real valued functions f on X with the property that sup{ f(x) ρ(x, x ) : x X} < for some chosen x X. For f, g C 1 (X), define σ(f, g) = sup X f g. 2. (a) Show that (C 1 (X), σ) is a complete metric space which is independent of the choice of x. (b) Define a map ι : X C 1 (X) by ι(x) = ρ x where ρ x denotes the function ρ x (y) = ρ(x, y). Show that ι is an isometric embedding of X into C 1 (X). (a) Let E n (, 1), n = 1,..., N, be measurable sets such that N n=1 m(e n) > N 1. Show that m( N n=1e n ) >. (b) Let E (, 1) be a measurable set with m(e) > 1 1/N. Let E denote the union of E with all of its integer translates E + k, k Z. Given any collection of points x 1,..., x N R, show that there is a point x (, 1) such that each x x n E for n = 1,..., N. 3. Let {f k } k=1 be an orthogonal set of functions in L2 (, 1) with f k (x) M for all k and almost every x (, 1). For n = 1, 2,... let σ n = 1 n n k=1 f k. (a) Show that k=1 σ k <. (b) Show that σ n (x) for almost every x (, 1). 4. Suppose that F R is a closed set without interior points. Construct a strictly increasing C 1 function f such that f (x) = if and only x F. 5. (a) Find an algebra of continuous complex-valued functions on the unit circle that separates points and contains the constants, but is not dense in the Banach space of continuous complex-valued functions with the usual maximum norm. Justify your answer. (Here an algebra means a complex vector space which is closed under multiplication.) (b) Let X be a compact Hausdorff space, and let C C (X) denote the Banach space of continuous complex-valued functions on X. Let A be a complex subalgebra of C C (X) that contains the constants, separates points, and is closed under complex conjugation (i.e. f A whenever f A). Show that A is dense. (You may assume the Stone-Weierstrass Theorem for real-valued functions.)

11 Do all five problems. Real Analysis Exam: Part I (Spring 2) 1. Let µ be a finite measure on R. Prove that for every x R, µ({x}) lim sup µ(ξ). ξ Here µ(ξ) = e ixξ dµ(x) denotes the Fourier transform of µ. 2. Let f : S 1 R be a Hölder continuous function on S 1 with Hölder exponent α. Thus f(x) f(y) sup x,y S 1, x y x y α <. Prove that for some constant C > depending on (the Hölder norm of) f but independent of n, the Fourier coefficient f(n) = 1 2π f(x)e inx dx satisfies f(n) C/ n α. S 1 3(a). Let E be a Banach space and E be its dual. Assume that E is uniformly convex. Prove that for every f E, there is one and only one g in the weak unit ball in E such that f E = f, g, where, is the pairing between E and E. (b). Suppose that f and g are two real-valued functions in L 3 (I), where I = [a, b], and that f 3 = g 3 = f(x) 2 g(x) dx = 1. Prove that g(x) = f(x) for almost every x I. You may do this either directly or else using part (a). 4. Let µ be a positive Borel measure supported in a compact set E R, with µ not identically zero. Assume that there are constants a > and C > such that for all intervals I, µ(i) C I a. (a) Define K b (x) = x b for b >. Prove that µ K b is well-defined and continuous for all b < a. (b) Give an explicit positive lower bound for the Hausdorff dimension of E. (Recall that the Hausdorff dimension of E is defined to be the supremum of values α such that the Hausdorff α -dimensional measure of E is infinite.) 5. Let T be a unitary operator on a Hilbert space H such that its spectrum σ(t ) is a countable set. Prove that there exists a sequence {n j } N such that T n j I (the identity operator) in the strong operator norm topology. (Hint: prove that there exists a sequence of exponentials {e injt } such that e in jt k 1 when e it k σ(t ). You may use Dirichlet s theorem that for any finite set {t k } N k=1 and for every ɛ >, there exists an n N such that e int k 1 < ɛ for k = 1,..., N.)

12 Do all five problems. Real Analysis Exam: Part II (Spring 2) 1. Let f C 2 (R) be a function such that M k = sup x R f (k) (x) < for k =, 1, 2. (a) Prove that M 1 2 M M 2. (Hint: use the second order Taylor formula for f(x+t) about t =.) (b) Show that the only functions for which equality is attained in part (a) are the constant functions. 2. Let m be Lebesgue outer measure on R and let µ be any outer measure on R. Suppose that all Borel sets in R are µ-measurable and, furthermore, that for every set A R, µ(a) is the infimum over all open sets U A of µ(u). If lim sup ρ µ([x ρ, x + ρ]) 2ρ 1 for all x R, prove that µ(a) m(a) for every subset A R. 3. Let H = H j be an orthonormal decomposition of the Hilbert space H into finitedimensional subspaces H j, and let {c j } be a sequence of positive numbers. The generalized cube determined by these data is the set Q = v H : v = v j with v j H j and v j c j. j=1 (a) Prove that the condition j=1 c2 j < is necessary and sufficient for the compactness of Q. (b) Prove that every compact set E H is contained in some compact generalized cube Q. 4. Let {A i } i=1 be a sequence of measurable subsets of a measure space (X, µ). Suppose that µ(a i ) <. Let Z = {x X : x A i for infinitely many i}. Prove that µ(z) =. 5. Assume that f L 1 (R) and that i=1 φ (x)f(x) dx 3 φ for every φ C (R). Prove that there is a Lipschitz continuous function g such that f(x) = g(x) for almost every x.

13 Real Analysis Qualifying Exam: Part I (Autumn 21) 1. Let K(x, y) C ([, 1] [, 1]). Prove that the mapping g(x) f(x) = (Kg)(x) = is a compact mapping from C([, 1]) to C([, 1]). 2. Prove the Poisson summation formula: 1 K(x, y)g(y) dy n= f(x + 2πn) = 1 2π k= f(k)e ikx for all f in the Schwartz space Here f(ξ) = R f(x)e ixξ dx. S {f : (1 + x 2 ) m f (n) (x) C m,n for all m, n }. 3. Let p be a number with 1 p <. Assume that f and f n, n = 1, 2,... are functions in L p (R n ) (with respect to standard Lebesgue measure), and that f n f almost everywhere. Prove that f n f L p if and only if f n L p f L p. 4. Suppose that the real-valued function f(x) is nondecreasing on the interval [, 1]. Prove that there exists a sequence of continuous functions f n (x) such that f n f pointwise on this interval. 5. Suppose f L 1 ([, 1]) but f / L 2 ([, 1]). Prove that there exists a complete orthonormal basis φ n for L 2 ([, 1]) such that for each n, φ n is continuous and moreover 1 f(x)φ n (x) dx =. 6. Let λ n be an arbitrary discrete sequence in R. Define Prove that f C (R), and that exists. f(x) = lim T 1 2T n=1 T T e iλ nx n 2. f(x) dx 1

14 Real Analysis Qualifying Exam: Part II (Autumn 21) 1. Does there exist a function f C ([, 1]) such that 1 1 xf(x) dx = 1, and x n f(x) dx = for n =, 2, 3,...? 2. Let {U n } be an orthonormal basis for a Hilbert space H. Let {V n } H be such that V n U n 2 = S <. Show that the linear span of {V n } is a subspace of finite codimension. Prove in fact that when S < 1, then {V n } is a basis for H. 3. Let f n L p ([, 1]), f n p 1 and assume that f n (x) almost everywhere. Prove that f n weakly. 4. Let (X j, d j ) be metric spaces, j = 1, 2. Let f : (X 1, d 1 ) (X 2, d 2 ) be distance nondecreasing, i.e. d 2 (f(y), f(z)) d 1 (y, z) for all y, z X 1. Are either of the following two implications true? Prove or give a counterexample. (a) If (X 1, d 1 ) is complete, then (X 2, d 2 ) is complete. (b) If (X 2, d 2 ) is complete, then (X 1, d 1 ) is complete. 5. Assume f real-valued and measurable on a probability measure space (e.g., on [, 1] with Lebesgue measure) and write Φ(λ) = µ({x : f(x) < λ}). Prove that for any continuous function g on R, (a) g f is measurable. (b) g f is integrable if and only if g is integrable with respect to the measure dφ, and g f dµ = R g(λ) dφ(λ). 6. Prove that for almost all x [, 1], there are at most finitely many rational numbers with reduced form p/q such that q 2 and x p/q < 1/(q log q) 2. Hint: Consider intervals of length 2/(q log q) 2 centered at rational points p/q. 2

15 Do all five problems. 1. Let X be a metric space. Real Analysis Exam: Part I (Spring 21) (a) Suppose X is separable. Show that if G is an open cover of X, then G has a countable subcover. (b) (Converse of (a)). Suppose every open cover of X has a countable subcover. Prove that X is separable. 2. Let T be the set of real numbers x with the following property. For every k <, there exist integers N > k and a such that x a 1 1 N 2 N. (a) Prove that T is uncountable. (b) What is the Lebesgue measure of T? 3. Let X be a compact metric space. Let C(X) be the space of all continuous real-valued functions on X. Suppose F : C(X) R is a continuous map such that F (u + v) = F (u) + F (v), F (uv) = F (u)f (v), F (1) = 1. Prove that there is an x X such that F (u) = u(x) for every u C(X). 4. Prove: (a) The continuous image of a connected set is connected. (b) If X is compact, Y is Hausdorff, and f : X Y is one-to-one and continuous, then f 1 is continuous. (c) The product of two compact spaces is compact. 5. Let S be a subspace of C[, 1]. Suppose S is closed as a subspace of L 2 [, 1]. Prove: (a) S is a closed subspace of C[, 1]. (b) For f S, f 2 f M f 2. (c) For every y [, 1], there is a K y L 2 [, 1] such that for every f S. f(y) = 1 K y (x)f(x) dx

16 Real Analysis Exam: Part II (Spring 21) 1. Suppose f n (x) is a sequence of non-decreasing functions on [, 1] that converge pointwise to a continuous function g(x). Prove that the convergence is actually uniform on [, 1]. 2. Let A and B be closed linear subspaces of a Hilbert space H such that inf{ x y : x A, y B, x = y = 1} >. Prove that A + B = {x + y : x A, y B} is complete. 3. Let A be the space of Fourier transforms of functions in L 1 (R): A = { f : f L 1 (R)}. Let C (R) be the space of continuous functions f on R such that Prove lim f(x) =. x (a) A C (R). (Hint: use the open mapping theorem.) (b) A is a dense subset of C (R). 4. Let Q be the unit square in R 2. Consider functions f n L 1 (Q) such that (as n ) f n f almost everywhere in Q and f n Q Q f <. (a) Prove that A f n f for every measurable subset A of Q. A (b) Prove that f n f in L Let f and g be continuous periodic functions with period 1. Prove that lim f(x)g(nx) dx = f(x) dx g(x) dx. n

17 Real Analysis Exam: Part I (Fall 22) 1. Let G be an unbounded open set in (, ), and let D = {x : nx G for infinitely many natural numbers n}. Prove that D is dense in (, ). 2. Suppose S is a linear subspace of C[, 1] such that for all f S, f λ f 2 where f and f 2 are the L (or sup) norm and the L 2 norm, respectively. Let v 1, v 2,..., v n be orthonormal functions in S. (a) For every x, show that there are numbers a i (x) (1 i n) such that a k (x) 2 = 1 and such that ( ak (x)v k (x) = vk (x) 2) 1/2. (b) Show that v k (x) 2 λ 2. (c) Show that dim S λ Let f C 1 (R) and suppose that f(x + 1) = f(x) for all x. Prove that f 1 f(t) dt + 1 f (t) dt. 4. Let µ be a non-negative Borel measure on R 2 with µ(r 2 ) = 1. Suppose that every set consisting of a single point has µ measure. Show that for every λ (, 1), there is Borel set E such that µ(e) = λ. 5(a). Let f be a function in L 1 (R) such that f = f f, where denotes convolution. Prove that f = almost everywhere. 5(b). Find all functions f on T (the reals mod 1) such that f = f f.

18 Real Analysis Exam: Part II (Fall 22) 1. Let f : R R be a function that is continuous except at a countable set of points. Prove that there is a sequence g n of continuous functions such that g n (x) f(x) for all x. Hint: Use piecewise linear functions. 2. Let f : R R be a C function with compact support. Suppose there is an infinite set S of positive integers such that f (n) (x) n! for all n S and for all x R. Prove that f. 3. Let g : R R be a C 1 function such that g(x + 1) = g(x) for all x. Let f(x) = 2 k g(2 k x). k=1 Show that there is a number A < such that for all x, y with x y 1 2. f(x) f(y) A x y log x y Hint: Assume 2 n 1 x y 2 n and divide the series for f into two parts. 4(a). For every ɛ >, show that there is function f L 1 [, 1] such that: f(x), f(x) = on a set of measure 1 ɛ, and b a f(x) dx > for every interval < a < b < 1. 4(b). Show that there is an absolutely continuous, strictly increasing function h on [, 1] such that h (x) = on a set of measure 1 ɛ. 5. Let f C (R) be a function and let f(ξ) = f(x)e 2πixξ dx be its Fourier transform. Assume that f(ξ) = if ξ 1/2. Show that (i) f(ξ) = n f(n)e 2πinξ for ξ 1 2 and that (ii) f(x) = n f(n) sin π(x n). π(x n)

19 Do all five problems. Real Analysis Exam: Part I (Spring 22) 1. Let f : R R be any function. Let E be the set of points x such that the limits f +(x) = f(x + h) f(x) lim h>,h h, f (x) f(x + h) f(x) = lim h<,h h exist and are not equal. That is, E = {x : f +(x), f (x) exist and f +(x) f (x)}. Prove that E is countable. Hint: for any fixed λ, consider the x for which f +(x) > λ > f (x). 2. Consider functions f n, g n L 2 [, 1] such that f n f and g n g weakly in L 2. (a) Show that the L 2 norms of the f n are uniformly bounded. (b) Show by example that f n g n need not converge to fg in the weak star topology of L 1 (where L 1 is regarded as part of the dual space of C[, 1].) (c) Suppose h n (n = 1, 2,... ) and h are in L 1 [, 1], and that the L 1 norms of the h n are uniformly bounded. Show that h n h in the weak star topology of L 1 if and only if each Fourier coefficient of h n converges to the corresponding Fourier coefficient of h. (d) Suppose that f n and g n each have Fourier series of the form k c ke 2πikx. Prove that f n g n fg in the weak star topology of L Consider a C function f : R R with the following property: for every x, f (k) (x) = for some k. Let U = {x : f is equal to a polynomial in some neighborhood of x}. (a) Prove that U is a dense open set. (b) Prove that the complement of U contains no isolated points. Remark: one can prove that U must be all of R, i.e., that f must be a polynomial. 4. Prove the inequality a n n m m n m 2 π 2 n a n 2 for all sequences a n l 2 (Z). Also, show that π 2 cannot be replaced by any smaller constant. Hint: First show that x 1 2 has Fourier series n e 2πinx 2πin. 5. Let K(x, y) be an L 1 function on the unit square [, 1] [, 1]. Suppose for every continuous function f on [, 1], we have K(x, y)f(y) dy = for almost every x. Prove that K = almost everywhere. x

20 Real Analysis Exam: Part II (Spring 22) Do all five problems. 1. Show that there do not exist measurable sets A and B in R, each of positive measure, such that A (B r) = for all rational r. 2. Let f n : [, 1] R be a sequence of Lebesgue measurable functions. Let E be the set of x such that n f n(x) converges. Show that for every ɛ >, there is a set F and a k < such that (i) F is in the ring of sets generated by sets of the form f 1 i (A), where i k and A is a Borel set, and (ii) m(e F ) < ɛ. 3. Let < C < 1. Show that there are numbers δ N (depending on C) with the following properties: (i) If A k (1 k N) are measurable sets in [, 1] each with measure C, then for some pair i, j with i j. (ii) δ N as N. m(a i A j ) (1 δ N )C 2 Hint: Let f k be the characteristic function of A k, let F = f f k, and consider F Suppose the Banach space X is uniformly convex. That is, suppose for every ɛ >, there is an δ > with the following property: If x = y = 1 and x y > δ, then (x + y)/2 < 1 ɛ. Let f be a linear functional on X with norm 1. Prove that there is a unique point x X such that x = 1 and f(x) = (a) Let µ be a finite measure on R, and let ν be the measure given by dν(x) = e x2 dµ(x). Show that the Fourier transform of ν is the restriction to R of an entire holomorphic function F. (b) Express the nth derivative F n () of F at in terms of µ. (c) Show that the set S = {p(x) exp( x 2 ) : p a polynomial} is dense in C (R), the space of continuous functions f : R R such that lim x f(x) = (with the sup norm).

21 Mathematics Department Stanford University Real Analysis Qualifying Exam, Autumn 23, Paper 1 1. Let f n be a sequence of continuous functions on R satisfying f n (x) f n+1 (x) 1 for all x R and all n {1, 2,...}. Let f(x) = lim f n (x). (a) Show that for all x we have f(x) lim inf y x f(y). (b) Assume that f is continuous at a point x. Show that for all ε >, there exist δ, N so that f n (y) f n (x) < ε whenever y x < δ and n > N. 2. Suppose that f n : [, 1] [, ) are non-negative Lebesgue measurable functions with f n (x) a.e. x [, 1], and assume sup n 1 ϕ(f n(x)) dx 1 for some continuous function ϕ : [, ) [, ) such that lim t t 1 ϕ(t) =. Prove that 1 f n(t) dt. 3. Suppose f L 1 ([, 2π]) and ˆf(n) = 2π f(x)e inx dx, n =, ±1,.... Prove the following: (a) n = ˆf(n) 2 < f L 2 ([, 2π]). (b) n = n ˆf(n) < the L 1 class of f has a representative f which extends to all of R as a 2π-periodic C 1 function. 4. Suppose f L 1 (R) and ˆf(ξ) = R f(x)e ixξ dx. Prove: (a) ˆf C(R) with ˆf(ξ) as ξ. (b) If f has compact support, ˆf cannot have compact support unless f =. 5. Let f be an arbitrary real-valued function on [, 1], and for each x (, 1) define Df(x) = lim sup y x f(y) f(x) y x. (a) If β R and if S β = {x (, 1) : Df(x) > β}, prove that for each ε > pairwise disjoint subintervals [a 1, b 1 ],..., [a N, b N ] [, 1] such that m (S β \ ( N i=1[a i, b i ])) < ε and β(b i a i ) < f(b i ) f(a i ) for each i = 1,..., N. (Here m denotes Lebesgue outer measure.) (b) If f is increasing on [, 1] show that the result of (a) directly implies the fact that Df(x) < a.e. x (, 1). prove that

22 Mathematics Department Stanford University Real Analysis Qualifying Exam, Autumn 23, Paper 2 1. Let B be a Banach space and S a linear map from B to C([, 1]) such that v n in B Sv n pointwise a.e. on [, 1]. Prove that S is a bounded operator from B to C([, 1]), assuming that C([, 1]) is equipped with its usual sup norm. 2. Let B(X) denote the set of functions X R, where X is a given non-empty set. (a) Describe a topology T on B(X) such that pointwise convergence of f n to f on X is equivalent to convergence of f n to f with respect to the topology T whenever f, f 1, f 2... are given functions in B(X). (b) If X is uncountable, show that T (as in (a)) is not metrizable. Hint: Assume a metric d for B(X) exists and consider the sets {x X : d(δ x, ) > ε} where ε > and δ x = 1 at x and zero elsewhere. 3. Suppose (X, A, µ) is an arbitrary measure space, f j f (weak convergence in L 2 ), and g j g pointwise with g j and g j µ-measurable for each j. Prove X gf 2 dµ lim inf j X g jf 2 j dµ. Hint: First prove g j f j converges weakly to gf in the case when g j K for some fixed K. 4. Let X be a normed linear space. X is said to be uniformly convex if there is a strictly increasing continuous function η on [, ) with η() = such that 1 x+y 1 η( x y ) 2 for every x, y X with x = y = 1. If X is a uniformly convex Banach space, show that for any decreasing sequence {K n } n=1,2,3,... of nonempty closed convex subsets of {x X : x 1}, we have n=1k n. 5. Suppose f is AC on [, 1]. Prove: (a) A [, 1] Lebesgue measurable f(a) Lebesgue measurable. Hint: Start by showing that f(a) has measure zero if A has measure zero. (b) If f() = f(1) and ˆf(n) = 1 f(x)e 2πinx dx, then n ˆf() as n.

23 Mathematics Department Stanford University Real Analysis Qualifying Exam, Spring 23, Paper 1 1. Let f be a continuous function on the unit square Q [, 1] [, 1], and for s [, 1] let g(s) = max{f(s, t) : t [, 1]}. (a) Show that g is a continuous function on [, 1]. (b) Prove that if f(x) f(y) M x y for x, y Q, then g(s 1 ) g(s 2 ) M s 1 s 2 for s 1, s 2 [, 1]. (c) Give an example in which f is C 1 but g is not C Suppose X, d is a metric space without isolated points (i.e. no single point is an open set) such that every continuous function f : X [, 1] is uniformly continuous. Prove that X is compact. 3. Suppose X, Y are Banach spaces and T : X Y is linear. Prove that T is bounded in each of the following cases: (a) If there is a family F of real continuous linear functionals on Y such that f T is continuous for each f F and f F f 1 {} = {}. (b) If there are closed sets A 1, A 2,... X with n=1a n = X and with T (A n ) a bounded subset of Y for each n = 1, 2, Suppose T : X Y is a compact bounded linear operator between Banach spaces (T compact means that the image of each bounded set has compact closure). Prove that the adjoint transformation T : Y X (defined by T (f) = f T for f Y ) is also compact. 5. A sequence {ξ j } j=1,2,... [, 1] is said to be uniformly distributed in the interval [, 1] 1 n if lim n n j=1 f(ξ j) = 1 f(x) dx for each f C([, 1]) (i.e. 1 n n j=1 δ ξ j Lebesgue measure on [, 1] in the weak sense). Prove that {ξ j } j=1,2,... is uniformly distributed in [, 1] if lim n 1 n n j=1 e2πimξ j = for each integer m. Hint: First consider the case when f() = f(1). 1

24 Mathematics Department Stanford University Real Analysis Qualifying Exam, Spring 23, Paper 2 1. If X is a finite dimensional real vector space, prove that all norms on X are equivalent (i.e. for each pair of norms 1, 2 on X there is a constant C 1 such that C 1 x 1 x 2 C x 1 for every x X). 2. (a) Prove that a weakly compact subset of a normed space X is bounded. (b) In the Hilbert space L 2 ([, 1]), give an example of a countable closed bounded subset that is not weakly closed, and justify your answer. 3. Let µ be a finite positive Borel measure on (, 1). (a) Prove that there is an increasing function α on (, 1) such that (,1) f dµ = 1 f (t)α(t) dt for each f C 1 ((, 1)) with compact support. (b) In case µ is non-atomic (i.e. in case µ({x}) = for each point x (, 1)), prove that α as in (a) is unique up to an additive constant and is also continuous. 4. Prove that the following integrals converge to zero as n : (a) (b) n 1 x 1/2 (1 + n 2 x 2 ) 1/2 cos nx dx. n(1 x) 2 cos nx dx. (1 + nx)(log x) 2 5. Prove that if α (, 1) and if f(t) is any L 2 function on the circle with Fourier series ˆf(n)e int such that n N ˆf(n) N α for each N 1, then the L 2 class of f(t) has a Hölder continuous representative f (t) with exponent α (i.e. f (t 1 ) f (t 2 ) C t 1 t 2 α for each t 1, t 2 ). Hint: n N n ˆf(n) CN 1 α for each N 1, with C a constant depending only on α. (Prove this fact if you make use of it.) 2

25 Real Analysis Quals Fall 24 Part I. Answer all problems 1 Prove that if 1 p < and f L p (R), then the map τ f τ, where f τ (x) = f(x τ), is continuous from R into L p (R) (endowed with the norm topology). 2 Let ( X, B, µ ) be a measure space, µ finite or infinite. For a measurable real-valued f on ( X, B, µ ) and n Z write m n = m n (f) = µ({x: 2 n 1 f(x) < 2 n }). Give a condition, stated in terms of {m n }, which is necessary and sufficient for f L p, where 1 p <. 3 Let ( X, B, µ ) be a finite measure space, µ(x) = a. a. Prove that L p ( 1 X, B, µ ) L p ) ( X, B, µ for 1 p p 1 <, and for f L p ) 1( X, B, µ, f p a ( 1 1 ) p p 1 f p1 Hint: Reduce to, and prove, the case a = 1. b. Let ( X, B, µ ) be a finite measure space, f n L 2( X, B, µ ), f n L 2 1, f measurable, and f n f in measure. Prove: b.1. f L 2( X, B, µ ). b.2. f n f in the L p -norm for every p < 2. b.3. f n f weakly (in L 2 ). 4 Denote by C ([, 1]) the space of all the infinitely differentiable complexvalued functions on the interval [, 1]. Define convergence in C ([, 1]) as follows: f n f in C if f n (j) f (j) uniformly for every j N. a. Prove that this convergence is equivalent to convergence in the metric d(f, g) = j= 2 j (f g) (j) 1 + (f g) (j) b. Prove that the topology so defined cannot be defined by a norm. 5 Let f L 1 (R) and assume that ˆf(ξ) = e ixξ f(x)dx = for ξ > 1. Prove that f is equal a.e. to the restriction to R of an entire function f(z), z = x + iy, such that lim y e y f(x + iy) =. OCTOBER 9, 24

26 Real Analysis Quals Fall 24 Part II. Answer all problems 6 Let {v n } be a sequence of vectors in a Hilbert space H, and assume that v n v in the weak topology. a. Prove that v n = O(1). b. Prove also that v n v in norm if, and only if v n v. 7 Let (X, ρ) be a complete metric space. A set E X has the property of Baire if E = (G \ P 1 ) P 2, where G is assumed open and P j, j = 1, 2 meager (first category). We denote by PB the set of all such E. a. Prove: PB is the sigma algebra generated by all the open sets and all the meager sets in X. Hint: The boundary of a closed set is non-dense. b. If E is a second category Borel set then it is residual (has meager complement) on some nonempty open subset. c. If E R is a second category Borel set then E E = {x y : x, y E} contains an interval ( δ, δ), δ >. 8 a. Let B C 1 ([, 1]) be a subspace of dimension N + 1. Show that there exists f B such that sup x 1 f(x) = 1 and sup x 1 f (x) 2N. Hint: There is a nontrivial g B which vanishes on E = { 2j 1 2N }N j=1. b. Prove that a subspace B C 1 ([, 1]) which is closed under uniform convergence is finite dimensional. Hint: Show that f K f for some constant K and all f B. 9 We denote by H-dim E the Hausdorff { dimension of a (closed) set E R k, H that is the number α such that α (E) = for α < α, H α (E) = for α > α, where H α (E) denotes the Hausdorff measure in dimension α of the set E. a. Let F be a Lipschitz map from a compact set E R k into R l. Show that F (E) is closed and H-dim E H-dim F (E). b. Prove that H-dim(bdry(D)) 1 for every non-empty bounded open set D R 2. 1 Assume that α is an irrational multiple of π. Let µ be a finite Borel measure on T = R/2πZ and assume that the measure ν defined for all Borel sets A T by: ν(a) = µ(a α) µ(a) is absolutely continuous (with respect to Lebesgue). Prove that µ is absolutely continuous (Lebesgue). Hint: Write µ = µ ac + µ s, the decomposition of µ to its absolutely continuous and singular parts. OCTOBER 9, 24

27 Ph.D. Qualifying Exam, Real Analysis September 25, part I Do all the problems. 1 (Quickies) a. Let B denote the set of all Borel probability measures on [, 1]? What are the extreme points of this set? b. Suppose that B is a Banach space B, and its dual B is separable. Prove that B is separable. Is the converse true? (prove or give a counterexample). 2 a. Suppose that P(ξ 1,...,ξ n ) is a polynomial on R n such that for some constants C 1,C 2 >, P(ξ) C 1 ξ when ξ C 2. Let P( ) be the differential operator defined by replacing each ξ j by / x j. Suppose that P( )u = f in R n, that f C (Rn ), and that u L p (R n ) for some 1 p. Prove that u C (R n ). b. Prove that for every φ C (R), lim ɛ + φ(x) x + iɛ dx exists, and that moreover the value of this limit depends continuously on φ in some C k norm. 3 Show that if g L 1 (T), µ M(T) (a finite measure on T), and µ(x + απ) µ(x) = gdt, for some irrational α, then µ is absolutely continuous. 4 Let f C (R) (the space of infinitely differentiable functions on the line). Assume that for every x R, f (n) (x) = for at least one n. Prove that f is a polynomial. Hints: Use Baire s theorem to show that there exists a dense open set G such that the restriction of f to any of its interval components agrees (on that interval) with a polynomial (i.e., for some n, which may depend on the component, f (n) (x) = identically). Use the Baire category theorem again. 5 Convolution and smoothness: a. Let f,g L 2 (T). Prove that f g C(T). b. Assume f C k (T) and g C l (T). Prove that f g C k+l (T). c. Construct a function ψ C(T) such that ψ ψ ψ (k times) is not differentiable for any k.

28 Ph.D. Qualifying Exam, Real Analysis September 25, part II Do all the problems. 1 (Quickies) a. Describe a norm on R 3 such that the unit vectors (1,,), (,1,) and (,,1) have norm 1 while (1,1,1) < 1 1. Hint: Think in terms of the unit ball. b. Let f n (t) = j Z ˆf n (j)e ijt where ˆf n (j) j log j for j > 75, uniformly in n. Assume that for all j, lim n ˆfn (j) exists, and denote it c j. Prove that g = c j e ijt C (T) and that f n converges to g in the topology of C k (T) for every k >. 2 Prove that every measurable homomorphism ϕ of T = R mod 2πZ into the multiplicative group T = {z: z = 1} C is given by ϕ(t) = e int with n N. Hint: Prove, and then use, the fact that ϕ is continuous. 3 The Hardy Littlewood maximal function of a function f L 1 (R) is defined by: 1 M f (x) = sup h> 2h x+h x h f(t) dt. a. Proof that, for f, M f is not integrable, but is of weak-l 1 -type, that is µ ({x; M f (x) > λ}) c λ. b. Identify the function m f (x) = lim sup h + 1 2h x+h x h f(t) dt. 4 T = R/2πZ is the circle group. Let k L 1 (T) and let K be the integral operator on L 2 (T) defined by K : f 1 2π k(x t)f(t)dt. a. Prove that K is compact and normal (i.e. commutes with its adjoint). When is it actually self-adjoint?

29 b. What is the spectrum of K and what are the corresponding eigenfunctions and eigenvalues? c. If we replace T by R, then is the analogous operator on L 2 (R) (with k L 1 (R)) necessarily compact? 5 Suppose that for some p, 1 < p <, f n L p ([,1]) and f n p 1, uniformly in n. Assuming that f n (x) a.e.; prove that f n weakly in L p.

30 Ph.D. Qualifying Exam problems, Real Analysis June 25, part I. 1 (Quickies) a. Let (B, ) be a normed space, and A : B B an invertible linear transformation such that A n c for some constant c > and all n Z. Prove that there is an equivalent norm on B with respect to which A is an isometry. b. Let µ be a finite measure on [ 1, 1] and assume that x kn = for some integer k and all nonnegative integers n. Prove that if k is odd then µ =. What can you say when k is even? c. Prove that the space C(, 1) is not reflexive. Hint: Identify the dual space M(, 1) of C(, 1) and show that the dual of M(, 1) contains elements that can not be identified with elements of C(, 1). d. Prove that C(, 1) is not isomorphic and in particular not isometric to a uniformly convex Banach space. 2 Prove that a linear operator T on a Hilbert space H is compact if, and only if, it is the limit, in the norm topology of operators, of a sequence of operators of finite rank. 3 Suppose B is a Banach space and K B is a subset. Recall that its convex hull, ch(k), is the smallest convex subset of B which contains K. a. Prove that if K is compact, then the closure of ch(k) is compact as well. b. Show that the set of indicator functions {11 [τ,τ+ 1 5 ] (t) : τ T} is compact in L 1 (T), and its convex hull is not. 4 Let A j [, 1], for j = 1, 2,..., N be Lebesgue measurable, µ(a j ) 1. 2 Let < a < 1/2 and denote E a = {x : x A j for more than an values of j}. Prove that µ(e a ) 1 2a. 2(1 a) Show that the estimate µ(e a ) 1 2a 2(1 a) is best possible (if it is to apply to all N). Hint: Consider F = N 1 11 A j. 5 The Hausdorff-Young inequality on the line states that if 1 p 2 and 1/p + 1/q = 1, then f L p (R) implies ˆf L q (ˆR) f L p (R). (1) Prove that the converse is true: (1) implies 1/p + 1/q = 1 and 1 p 2. Hint: For the first claim use scaling (f λ,s = λ s f(λx) for appropriate s, and its effect on the Fourier transform). For the second claim, show that if ϕ λ (x) = e iλx2 11 [ 1,1], then as λ ˆϕ λ while ˆϕ λ 2 remains constant.

31 Ph.D. Qualifying Exam problems, Real Analysis June 25, part II. 6 (Quickies) a. A distribution µ on T is positive if f, µ for every nonnegative f C (T). Show that a positive distribution is a measure. Hint: Positivity implies: for real-valued f, f, µ max f(t) 1, µ. b. Assume f n L 2 [, 1], n N, and f n 1. Prove that µ({x f n (x) > n 2 3 }) < n 4 3, and conclude that for every ε > there exists N N such that the measure of the set {x : f n (x) n 2 3 for n > N} exceeds > 1 ε. 7 Let {f n } be an orthonormal sequence in L 2 (, 1). Prove that S n = 1 n fm a.e. Hints: a. S n L 2 = 1. It follows that if n λ 1 j <, and in particular if λ j = [j log 2 j], then Sλj 2 converges a.e. and S λj a.e. b. If N (λ j, λ j+1 ), then S N = λ j N S λ j + 1 N N λ j +1 f m. Use 6.b. to estimate the last sum. 8 Let B = {(x, y, z) : r = x 2 + y 2 + z 2 1}, the unit ball in R 3, G = 11 B its indicator function, and g(x) = G(x, y, z) dy dz. a. Compute g. b. Show that ĝ(ξ) = O ( ξ 3) ) c. Prove ((ξ Ĝ(ξ, η, ζ) = O 2 + η 2 + ζ 2 ) 3 2. d. Let F n (x, y, z) = sin 2 (nr)g(x, y, z), and f n (x) = F n (x, y, z)dy dz. Prove: f n 1 2g uniformly as n, 9 Let f(x) = 1 n cos 1 2n x a. Prove that f satisfies the Hölder 1 2 condition: f(x + h) f(x) const h 1 2. b. Prove that f is nowhere differentiable. Hint: For every x and every n find points y n and z n such that x y n 1 2n, x z n 1 2n f(x) f(y, n ) x y n > 1 n 2, and f(x) f(y n) x y n < 1 n 2. c. Can a Lipschitz function be nowhere differentiable? (Justify your answer by quoting relevant standard theorems.) 1 Let B be a Banach space and S a linear map from B into C([, 1]), such that if {v n } B and v n B then Sv n (x) pointwise in [, 1]. Prove that S is bounded; in particular, the assumptions v n B implies Sv n (x) uniformly.

32 Ph.D. Qualifying Exam, Real Analysis September 26, part I Do all the problems. 1 Let G be an unbounded open set in (, ). Define D = {x (, ) : nx G for infinitely many n}. Prove that D is dense in (, ). 2 Suppose that f L 1 ([, 1]) but f / L 2 ([, 1]). Find a complete orthonormal basis {φ n } for L 2 ([, 1]) such that each φ n C ([, 1]) and such that 1 f(x)φ n (x) dx = n. 3 Let H 1 and H 2 be two separable Hilbert spaces. Suppose that A : H 1 H 2 is a continuous injective linear map. Suppose that {v j } is a bounded sequence in H 1 such that Av j converges strongly in H 2 to some element w. Prove that there exists an element v H 1 such that v j converges weakly to v and Av = w. 4 Some problems about linear functionals: a. Let T : C ([, 1]) C be defined by T (f) = f(1/2). Is T continuous with respect to the L 2 norm? Explain why or why not. b. Let P denote the set of polynomials of arbitrary degree, and consider their restrictions to [, 1] so as to consider P C ([, 1]). Define the linear functional on P, T k (f) = a k where a k is the coefficient of x k. Does T k extend as a continuous linear functional to all of C ([, 1])? Hint: Consider f n (x) = (1 x) n, n k. 5 Let Q = [, 1] [, 1] and denote by X be the set of all closed nonempty subsets of Q. Define d(a, B) = inf{δ > : A B δ and B A δ }, where for any C X, C δ = {x Q : dist (x, C) < δ}. Prove that (X, d) is a compact metric space. Hints: First prove that the subset of elements A X where A is finite is dense in X. Next, if {A n } is a decreasing nested sequence of closed subsets of Q, A n = A, prove that A n A. Finally, if {B n } is an arbitrary Cauchy sequence in (X, d), consider A n = k n A k.

33 Ph.D. Qualifying Exam, Real Analysis September 26, part II Do all the problems. 1 Let H be a Hilbert space with an orthogonal decomposition into finite dimensional subspaces, H = j=1 H j. Thus each v H can be written uniquely as v = j=1 v j with v j H j. Let c = (c 1, c 2,...) where each c j >, and define the subset A c = {v : v j c j } H. a. Prove that c l 2 if and only if A c is compact in H. b. Prove that every compact subset K H is contained in some A c for some c l 2. 2 Let µ be a finite measure on R. Define its Fourier transform µ(ξ) = e ixξ dµ(x). Prove that µ({x}) lim sup µ(ξ). ξ 3 Suppose that p, q, r [1, ) satisfy 1 p + 1 q = r. Prove that for every f Lp (R) and g L 1 (R), their convolution is an element of L r (R), and that ( ) r f g r dx f p p ) r dx g q q dx. Hint: Use interpolation. 4 Let g : R R be a C 1 function such that g(x + 1) = g(x) for every x. Define f(x) = 2 k g(2 k x). k=1 Show that there exists A > such that for all x, y R, f(x) f(y) A x y log x y. Hint: If 2 n 1 x y 2 n, divide the series for f into two parts. 5 Suppose that f is a function from the natural numbers N to R + such that for every value of n, m N, f(n + m) f(n) + f(m). Prove that exists and equals f(n) lim n n f(n) inf n> n.

34 Ph.D. Qualifying Examination, Real Analysis Spring 26, part I Do all the problems. 1 Quickies a. Let f n L p ([,1]) where 1 < p <. Suppose that f n p 1 and moreover that f n (x) for a.e. x. Prove that f n weakly in L p. b. Let v 1,...,v N be a finite sequence of unit vectors in a Hilbert space H. Suppose that there exists a number a (, 1) such that v i,v j a, i j. Find an upper bound for N in terms of a. c. Let h L 2 (S 1 ) and assume that h(t) for a.e. t S 1. Prove that the subspace is dense. V = {P(t)h(t) : P(t) a trigonometric polynomial} L 2 (S 1 ) 2 Let {f k } be a sequence of real-valued functions defined on [ 1,1] such that f k (x) f k (y) x y + 1 k for all k and x,y [ 1,1]. Suppose also that each f k () =. Prove that some subsequence of the f k converges uniformly to a continuous function f on [ 1,1]. 3 a. Construct a sequence {f n } of positive continuous functions on R such that f n (x) is bounded as n when x Q, but f n (x) is unbounded for x R \ Q. b. Prove that there is no sequence {g n } of positive continuous functions such that g n (x) is bounded when x R \ Q, but g n (x) is unbounded when x Q. 4 Let A R be a Lebesgue measurable set with < µ(a) < (µ is Lebesgue measure). Let µ(a [a r,a + r]) d(x,r) =. 2r Prove that there is a point x R such that < lim inf r d(x,r) lim sup d(x,r) < 1. r 5 Let C be a closed convex set in a Hilbert space H. Prove that C contains a unique element of minimal norm.