Part I 240B Homework Exercises (Winter 2012)bdriver/240b-c_2012/Homeworks/240B_Home… · 10 3 HomeWork #3 Due Friday January 27, 2012 1.If fx ngis Cauchy sequence, then there is

Part I

240B Homework Exercises (Winter 2012)

3

The problems below either come from the lecture notes or from Folland. At the moment in the lecturenotes please use the following translation table for this course:

a) Random variable = Measurable function on some measure space.b) Probability measure is a measure where the total measure of the space is 1.c) E [X] or EX is short hand for ∫

Ω

X (ω) dµ (ω) ,

i.e. it is simply shorthand notation for the integral of X relative to a measure, in this case the measureis assumed to µ and the underlying set is Ω.

1

HomeWork # 1 Due Friday January 13, 2012

Exercise 1.1 (Measurability). Please review or read Lemma 9.3 of the lecture notes. I will likely use thislemma without mention throughout the course.

Exercise 1.2 (Differentiating past the integral). Please review (or read if it is new to you) Corollary10.30, Corollary 10.31, and Proposition 10.33 in the lecture notes (Ver. 1). Also review Section 10.3 of thelecture notes (Ver. 1).

Exercise 1.3 (Folland 2.28 on p. 60.). Compute the following limits and justify your calculations:

1. limn→∞

∫∞0

sin( xn )

(1+ xn )n dx.

2. limn→∞

∫ 1

01+nx2

(1+x2)n dx

3. limn→∞

∫∞0

n sin(x/n)x(1+x2) dx

4. For all a ∈ R compute,

f (a) := limn→∞

∫ ∞a

n(1 + n2x2)−1dx.

[Hints: for parts 1. and 2. you might use the binomial expansion to estimate the denominators.]

Exercise 1.4. Let Ω := 1, 2, 3, 4 and M := 1A, 1B where A := 1, 2 and B := 2, 3 .

a) Show σ (M) = 2Ω .b) Find two distinct probability measures, µ and ν on 2Ω such that µ (A) = ν (A) and µ (B) = ν (B) , i.e.∫

Ω

fdµ =

∫Ω

fdν (1.1)

holds for all f ∈M.

Moral: the assumption that M is multiplicative can not be dropped from multiplicative system theorem.

Exercise 1.5. Suppose that u : R→ R is a continuously differentiable function such that u (t) ≥ 0 for all tand limt→±∞ u (t) = ±∞. Show that ∫

Rf (x) dx =

∫Rf (u (t)) u (t) dt (1.2)

for all measurable functions f : R→ [0,∞] . In particular applying this result to u (t) = at + b where a > 0implies, ∫

Rf (x) dx = a

∫Rf (at+ b) dt.

6 1 HomeWork # 1 Due Friday January 13, 2012

Exercise 1.6. Let (Ω,B, P ) be a probability space and X,Y : Ω → R be a pair of random variables suchthat

E [f (X) g (Y )] = E [f (X) g (X)]

for every pair of bounded measurable functions, f, g : R→ R. Show P (X = Y ) = 1. Hint: Let H denotethe bounded Borel measurable functions, h : R2 → R such that

E [h (X,Y )] = E [h (X,X)] .

Use the multiplicative systems Theorem ?? to show H is the vector space of all bounded Borel measurablefunctions. Then take h (x, y) = 1x=y.

Exercise 1.7 (Density of A – simple functions). Let (Ω,B, P ) be a probability space and assume thatA is a sub-algebra of B such that B = σ (A) . Let H denote the bounded measurable functions f : Ω → Rsuch that for every ε > 0 there exists an an A – simple function1, ϕ : Ω → R such that E |f − ϕ| < ε. ShowH consists of all bounded measurable functions, f : Ω → R. Hint: let M denote the collection of A – simplefunctions.

Exercise 1.8 (Density of A in B = σ (A) – NOT to be collected). Keeping the same notation as inExercise 1.7 but now take f = 1B for some B ∈ B and given ε > 0, write ϕ =

∑ni=0 λi1Ai

where λ0 = 0,λini=1 is an enumeration of ϕ (Ω) \ 0 , and Ai := ϕ = λi . Show; 1.

E |1B − ϕ| = P (A0 ∩B) +

n∑i=1

[|1− λi|P (B ∩Ai) + |λi|P (Ai \B)] (1.3)

≥ P (A0 ∩B) +

n∑i=1

min P (B ∩Ai) , P (Ai \B) . (1.4)

2. Now let ψ =∑ni=0 αi1Ai

with

αi =

1 if P (Ai \B) ≤ P (B ∩Ai)0 if P (Ai \B) > P (B ∩Ai)

.

Then show that

E |1B − ψ| = P (A0 ∩B) +

n∑i=1

min P (B ∩Ai) , P (Ai \B) ≤ E |1B − ϕ| .

Observe that ψ = 1D where D = ∪i:αi=1Ai ∈ A and so you have shown; for every ε > 0 there exists a

D ∈ A such thatP (B∆D) = E |1B − 1D| < ε.

[This problem is the same as one of the propositions in the notes!]

Exercise 1.9. Suppose that (Xi,Bi)ni=1 are measurable spaces and for each i,Mi is a multiplicative systemof real bounded measurable functions on Xi such that σ (Mi) = Bi and there exist χn ∈Mi such that χn → 1boundedly as n→∞. Given fi : Xi → R let f1 ⊗ · · · ⊗ fn : X1 × · · · ×Xn → R be defined by

(f1 ⊗ · · · ⊗ fn) (x1, . . . , xn) = f1 (x1) . . . fn (xn) .

ShowM1 ⊗ · · · ⊗Mn := f1 ⊗ · · · ⊗ fn : fi ∈Mi for 1 ≤ i ≤ n

is a multiplicative system of bounded measurable functions on

(X := X1 × · · · ×Xn,B := B1 ⊗ · · · ⊗ Bn)

such that σ (M1 ⊗ · · · ⊗Mn) = B. (It is enough to write your solution in the special case where n = 2.)

1 Recall from Definition ?? than f is an A – simple function if f is a simple function such that f−1 (y) ∈ A forall y ∈ R.

2

HomeWork # 2 Due Friday January 20, 2012

Exercise 2.1. Folland 3.27 on p. 107.

Exercise 2.2 (Look at but do not hand in). Folland 3.28 on p. 107.

Exercise 2.3 (Look at but do not hand in). Folland 3.29 on p. 107.






Exercise 2.9 (Look at but do not hand in.). Folland 3.40 on p. 108.

3

HomeWork #3 Due Friday January 27, 2012

Exercise 3.1 (Global Integration by Parts Formula). Suppose that f, g : R→ R are locally absolutelycontinuous functions1 such that f ′g, fg′, and fg are all Lebesgue integrable functions on R. Prove thefollowing integration by parts formula;∫

Rf ′ (x) · g (x) dx = −

∫Rf (x) · g′ (x) dx. (3.1)

Similarly show that; if f, g : [0,∞)→ [0,∞) are locally absolutely continuous functions such that f ′g, fg′,and fg are all Lebesgue integrable functions on [0,∞), then∫ ∞

0

f ′ (x) · g (x) dx = −f (0) g (0)−∫ ∞0

f (x) · g′ (x) dx. (3.2)

Outline: 1. First use the theory developed to see that Eq. (3.1) holds if f (x) = 0 for |x| ≥ N for someN <∞.

2. Let ψ : R→ [0, 1] be a continuously differentiable function such that ψ (x) = 1 if |x| ≤ 1 and ψ (x) = 0if |x| ≥ 2.2 For any ε > 0 let ψε(x) = ψ(εx) Write out the identity in Eq. (3.1) with f (x) being replaced byf (x)ψε (x) .

3. Now use the dominated convergence theorem to pass to the limit as ε ↓ 0 in the identity you found instep 2.

4. A similar outline works to prove Eq. (3.2).

Exercise 3.2. Let (X, d) be a metric space. Suppose that xn∞n=1 ⊂ X is a sequence and set εn :=d(xn, xn+1). Show that for m > n that

d(xn, xm) ≤m−1∑k=n

εk ≤∞∑k=n

εk.

Conclude from this that if∞∑k=1

εk =

∞∑n=1

d(xn, xn+1) <∞

then xn∞n=1 is Cauchy. Moreover, show that if xn∞n=1 is a convergent sequence and x = limn→∞ xn then

d(x, xn) ≤∞∑k=n

εk.

Exercise 3.3. Show that (X, d) is a complete metric space iff every sequence xn∞n=1 ⊂ X such that∑∞n=1 d(xn, xn+1) <∞ is a convergent sequence inX. You may find it useful to prove the following statements

in the course of the proof.

1 This means that f and g restricted to any bounded interval in R are absolutely continuous on that interval.2 You may assume the existence of such a ψ, we will deal with this later.

10 3 HomeWork #3 Due Friday January 27, 2012

1. If xn is Cauchy sequence, then there is a subsequence yj := xnj such that∑∞j=1 d(yj+1, yj) <∞.

2. If xn∞n=1 is Cauchy and there exists a subsequence yj := xnjof xn such that x = limj→∞ yj exists,

then limn→∞ xn also exists and is equal to x.

Exercise 3.4. Suppose that f : [0,∞) → [0,∞) is a C2 – function such that f(0) = 0, f ′ > 0 and f ′′ ≤ 0and (X, ρ) is a metric space. Show that d(x, y) = f(ρ(x, y)) is a metric on X. In particular show that

d(x, y) :=ρ(x, y)

1 + ρ(x, y)

is a metric on X. (Hint: use calculus to verify that f(a+ b) ≤ f(a) + f(b) for all a, b ∈ [0,∞).)

Exercise 3.5. Let (Xn, dn)∞n=1 be a sequence of metric spaces, X :=∏∞n=1Xn, and for x = (x(n))

∞n=1

and y = (y(n))∞n=1 in X let

d(x, y) =

∞∑n=1

2−ndn(x(n), y(n))

1 + dn(x(n), y(n)). (3.3)

Show:

1. (X, d) is a metric space,2. a sequence xk∞k=1 ⊂ X converges to x ∈ X iff xk(n)→ x(n) ∈ Xn as k →∞ for each n ∈ N and3. X is complete if Xn is complete for all n.

Exercise 3.6 (Look at but do not hand in). Let (X, ‖·‖) be a normed space over F (R or C). Show themap

(λ, x, y) ∈ F×X ×X → x+ λy ∈ X

is continuous relative to the norm on F×X ×X defined by

‖(λ, x, y)‖F×X×X := |λ|+ ‖x‖+ ‖y‖ .

Exercise 3.7. Let X = N and for p, q ∈ [1,∞) let ‖·‖p denote the `p(N) – norm. Show ‖·‖p and ‖·‖q areinequivalent norms for p 6= q by showing

supf 6=0

‖f‖p‖f‖q

=∞ if p < q.

Exercise 3.8 (Look at but do not hand in.). Let d : C(R)× C(R)→ [0,∞) be defined by

d(f, g) =

∞∑n=1

2−n‖f − g‖n

1 + ‖f − g‖n,

where ‖f‖n := sup|f(x)| : |x| ≤ n = max|f(x)| : |x| ≤ n.

1. Show that d is a metric on C(R).2. Show that a sequence fn∞n=1 ⊂ C(R) converges to f ∈ C(R) as n→∞ iff fn converges to f uniformly

on bounded subsets of R.3. Show that (C(R), d) is a complete metric space.

[The solution to this exercise is similar to the solution to Exercise 3.5.]

Exercise 3.9. Let X = C([0, 1],R) and for f ∈ X, let

‖f‖1 :=

∫ 1

0

|f(t)| dt.

Show that (X, ‖·‖1) is normed space and show by example that this space is not complete.

3 HomeWork #3 Due Friday January 27, 2012 11

Exercise 3.10. By making the change of variables, u = lnx, prove the following facts:∫ 1/2

0

x−a |lnx|b dx <∞⇐⇒ a < 1 or a = 1 and b < −1∫ ∞2

x−a |lnx|b dx <∞⇐⇒ a > 1 or a = 1 and b < −1∫ 1

0

x−a |lnx|b dx <∞⇐⇒ a < 1 and b > −1∫ ∞1

x−a |lnx|b dx <∞⇐⇒ a > 1 and b > −1.

Suppose 0 < p0 < p1 ≤ ∞ and m is Lebesgue measure on (0,∞) . Use the above results to manufacture afunction f on (0,∞) such that f ∈ Lp ((0,∞) ,m) iff (a) p ∈ (p0, p1) , (b) p ∈ [p0, p1] and (c) p = p0.

Exercise 3.11. Let (X,B, µ) be a measure space and g ∈ L1 (µ) . Show that∣∣∫Xgdµ

∣∣ = ‖g‖1 iff there existsz ∈ C with |z| = 1 such that g = |g| z a.e. (This may be equivalently stated as sgn(g (x)) := g (x) / |g (x)| isconstant for µ -a.e. on the set where g 6= 0.) In particular for f ∈ Lp (µ) and g ∈ Lq (µ) with q = p/ (p− 1) ,we have, ∣∣∣∣∫

X

fgdµ

∣∣∣∣ =

∫X

|fg| dµ ⇐⇒ sgn(f (x)) = sgn(g (x)) for µ–a.e. x ∈ fg 6= 0 .

Exercise 3.12. Let (X, ‖·‖1) be the normed space in Exercise 3.9. Compute the closure of A when

1. A = f ∈ X : f (1/2) = 0 .2. A =

f ∈ X : supt∈[0,1] f (t) ≤ 5

. [Hint: you may use without proof that if fn ∈ A converges to f ∈ X

in ‖·‖1 then there is a subsequence which converges for a.e. t.]

3. A =f ∈ X :

∫ 1/2

0f (t) dt = 0

.

4

HomeWork #4 Due Friday February 3, 2012

Definition 4.1. Given a set A contained in a metric space X, let A ⊂ X be the closure of A defined by

A := x ∈ X : ∃ xn ⊂ A 3 x = limn→∞

xn.

That is to say A contains all limit points of A. We say A is dense in X if A = X, i.e. every elementx ∈ X is a limit of a sequence of elements from A. A metric space is said to be separable if it contains acountable dense subset, D.

Exercise 4.1. Given A ⊂ X, show A is a closed set and in fact

A = ∩F : A ⊂ F ⊂ X with F closed. (4.1)

That is to say A is the smallest closed set containing A.

Exercise 4.2. If D is a dense subset of a metric space (X, d) and E ⊂ X is a subset such that to everypoint x ∈ D there exists xn∞n=1 ⊂ E with x = limn→∞ xn, then E is also a dense subset of X. If points inE well approximate every point in D and the points in D well approximate the points in X, then the pointsin E also well approximate all points in X.

Exercise 4.3 (Look at but do not hand in). Suppose that (X, ‖·‖) is a normed space and S ⊂ X is alinear subspace.

1. Show the closure S of S is also a linear subspace.2. Now suppose that X is a Banach space. Show that S with the inherited norm from X is a Banach space

iff S is closed.

Exercise 4.4. Let Y = BC (R,C) be the Banach space of continuous bounded complex valued functions onR equipped with the uniform norm, ‖f‖u := supx∈R |f (x)| . Further let C0 (R,C) denote those f ∈ C (R,C)such that vanish at infinity, i.e. limx→±∞ f (x) = 0. Also let Cc (R,C) denote those f ∈ C (R,C) withcompact support, i.e. there exists N <∞ such that f (x) = 0 if |x| ≥ N. Show C0 (R,C) is a closed subspaceof Y and that

Cc (R,C) = C0 (R,C) .

Exercise 4.5. Let (X, ‖·‖) be a Banach space. To each A ∈ L (X) , we may define LA, RA : L (X)→ L (X)by

LAB = AB and RAB = BA for all B ∈ L (X) .

Show LA, RA ∈ L (L (X)) and that

‖LA‖L(L(X)) = ‖A‖L(X) = ‖RA‖L(L(X)) .

For the next three exercises, let X = Rn and Y = Rm and T : X → Y be a linear transformation so thatT is given by matrix multiplication by an m × n matrix. Let us identify the linear transformation T withthis matrix.

14 4 HomeWork #4 Due Friday February 3, 2012

Exercise 4.6. Assume the norms on X and Y are the `1 – norms, i.e. for x ∈ Rn, ‖x‖ =∑nj=1 |xj | . Then

the operator norm of T is given by

‖T‖ = max1≤j≤n

m∑i=1

|Tij | .

Exercise 4.7. Assume the norms on X and Y are the `∞ – norms, i.e. for x ∈ Rn, ‖x‖ = max1≤j≤n |xj | .Then the operator norm of T is given by

‖T‖ = max1≤i≤m

n∑j=1

|Tij | .

Exercise 4.8. Assume the norms on X and Y are the `2 – norms, i.e. for x ∈ Rn, ‖x‖2 =∑nj=1 x

2j . Show

‖T‖2 is the largest eigenvalue of the matrix T trT : Rn → Rn. Hint: Use the spectral theorem for symmetricreal matrices.

Lemma 4.2. Let X, Y and Z be normed spaces. Then the maps

(S, x) ∈ L(X,Y )×X −→ Sx ∈ Y

and(S, T ) ∈ L(X,Y )× L(Y, Z) −→ TS ∈ L(X,Z)

are continuous relative to the norms

‖(S, x)‖L(X,Y )×X := ‖S‖L(X,Y ) + ‖x‖X and

‖(S, T )‖L(X,Y )×L(Y,Z) := ‖S‖L(X,Y ) + ‖T‖L(Y,Z)

on L(X,Y )×X and L(X,Y )× L(Y, Z) respectively.

Exercise 4.9. Suppose that X,Y and Z are Banach spaces and Q : X × Y → Z is a bilinear form, i.e. weare assuming x ∈ X → Q (x, y) ∈ Z is linear for each y ∈ Y and y ∈ Y → Q (x, y) ∈ Z is linear for eachx ∈ X. Show Q is continuous relative to the product norm, ‖(x, y)‖X×Y := ‖x‖X + ‖y‖Y , on X×Y iff thereis a constant M <∞ such that

‖Q (x, y)‖Z ≤M ‖x‖X · ‖y‖Y for all (x, y) ∈ X × Y. (4.2)

Then apply this result to prove Lemma 4.2.

Exercise 4.10. Let (X,M, µ) and (Y,N , ν) be σ – finite measure spaces, f ∈ L2(ν) and k ∈ L2(µ ⊗ ν).Show ∫

|k(x, y)f(y)| dν(y) <∞ for µ – a.e. x (4.3)

and define

Kf(x) :=

∫Y

k(x, y)f(y)dν(y) (4.4)

when the integral is defined and set Kf (x) = 0 otherwise. Show Kf ∈ L2(µ) and K : L2(ν) → L2(µ) isa bounded operator with ‖K‖op ≤ ‖k‖L2(µ⊗ν) . [Remember that L2 (ν) and L2 (µ) consists of equivalence

classes of functions.]

Theorem 4.3 (B. L. T. Theorem). Suppose that Z is a normed space, X is a Banach space, and S ⊂ Zis a dense linear subspace of Z. If T : S → X is a bounded linear transformation (i.e. there exists C < ∞such that ‖Tz‖ ≤ C ‖z‖ for all z ∈ S), then T has a unique extension to an element T ∈ L(Z,X) and thisextension still satisfies ∥∥T z∥∥ ≤ C ‖z‖ for all z ∈ S.

Exercise 4.11 (Look at but do not hand in). Prove Theorem 4.3.

5


Exercise 5.1. Suppose (X, d) is a metric space which contains an uncountable subset Λ ⊂ X with theproperty that there exists ε > 0 such that d (a, b) ≥ ε for all a, b ∈ Λ with a 6= b. Show that (X, d) is notseparable.

Exercise 5.2. Show `∞ (N) is not separable. Hint: find an uncountable set Λ ⊂ `∞ (N) as in the hypothesisof Exercise 5.1.

Exercise 5.3. Suppose M is a subset of H, then M⊥⊥ = span(M) where (as usual), span (M) denotes allfinite linear combinations of elements from M.

Exercise 5.4. Let H,K,M be Hilbert spaces, A,B ∈ L(H,K), C ∈ L(K,M) and λ ∈ C. Show (A+ λB)∗

=A∗ + λB∗ and (CA)

∗= A∗C∗ ∈ L(M,H).

Exercise 5.5. Let H = Cn and K = Cm equipped with the usual inner products, i.e. 〈z|w〉H = z · w forz, w ∈ H. Let A be an m×n matrix thought of as a linear operator from H to K. Show the matrix associatedto A∗ : K → H is the conjugate transpose of A.

Remark 5.1 (Polarization identity). It is sometimes useful to know that the inner product 〈·|·〉 may bereconstructed from knowledge of its associated norm ‖·‖ . For example in the real case we have

〈x|y〉 =1

2

(‖x+ y‖2 − ‖x‖2 − ‖y‖2

). (5.1)

Similarly if we are working over C, then by direct computation,

2 Re〈x|y〉 = ‖x+ y‖2 − ‖x‖2 − ‖y‖2

and−2Re〈x|y〉 = ‖x− y‖2 − ‖x‖2 − ‖y‖2.

Subtracting these two equations gives the “polarization identity,”

4Re〈x|y〉 = ‖x+ y‖2 − ‖x− y‖2.

Replacing y by iy in this equation then implies that

4Im〈x|y〉 = ‖x+ iy‖2 − ‖x− iy‖2

from which we find

〈x|y〉 =1

4

∑ε∈G

ε‖x+ εy‖2 (5.2)

where G = ±1,±i – a cyclic subgroup of S1 ⊂ C.


Exercise 5.6. Let U : H → K be a linear map between complex Hilbert spaces. Show the following areequivalent:

1. U : H → K is an isometry,2. 〈Ux|Ux′〉K = 〈x|x′〉H for all x, x′ ∈ H,3. U∗U = idH .

Hint: use the polarization identity in Eq. (5.2) of Remark 5.1.

Exercise 5.7. Let U : H → K be a linear map, show the following are equivalent:

1. U : H → K is unitary2. U∗U = idH and UU∗ = idK .3. U is invertible and U−1 = U∗.

Definition 5.2 (Strong Convergence). Let X be a Banach space. We say a sequence of operatorsAn∞n=1 ⊂ L (X) converges strongly to A ∈ L (X) if limn→∞Anx = Ax for all x ∈ X. We abbrevi-ate this by [Note well that strong convergence is weaker than norm convergence, i.e. if ‖A−An‖op → 0

then Ans→ A as n→∞ but no necessarily the other way around unless dimX <∞.]

Exercise 5.8 (Strong convergence of increasing projections). Let (H, 〈·|·〉) be a Hilbert space andsuppose that Pn∞n=1 is a sequence of orthogonal projection operators on H such that Pn(H) ⊂ Pn+1(H)for all n. Let M := ∪∞n=1Pn(H) (a subspace of H) and let P denote orthonormal projection onto M. Show

Pns→ P as n → ∞, i.e. limn→∞ Pnx = Px for all x ∈ H. Hint: first prove the result for x ∈ M⊥, then for

x ∈M and then for x ∈ M.

Exercise 5.9 (The Mean Ergodic Theorem). Let U : H → H be a unitary operator on a Hilbert space

H, M = Nul(U − I), P = PM be orthogonal projection onto M, and Sn = 1n

∑n−1k=0 U

k. Show Sns→ PM as

n→∞, i.e. limn→∞ Snx = PMx for all x ∈ H.Hints: 1. Show H is the orthogonal direct sum of M and Ran(U − I) by using a Lemma from the notes

after first showing Nul(U∗ − I) = Nul(U − I). 2. Verify the result for x ∈ Nul(U − I) and x ∈ Ran(U − I).3. Use a limiting argument to verify the result for x ∈ Ran(U − I).

5.1 Extra problems to ponder – do not hand in. 17

5.1 Extra problems to ponder – do not hand in.

Exercise 5.10. Let (X,B, µ) be a σ – finite measure space. Suppose that 1 ≤ p <∞ and to each ϕ ∈ L∞ (µ)show Mϕ ∈ L (Lp (µ)) be defined by Mϕf = ϕf for all f ∈ Lp (µ) – so Mϕ is multiplication by ϕ. Show‖Mϕ‖op = ‖ϕ‖∞ , i.e. L∞ (µ) 3 ϕ→ Mϕ ∈ L (Lp (µ)) is an isometry.

Exercise 5.11. Let m be Lebesgue measure on ([0,∞),B,m) .

1. Show L∞ (([0,∞),B,m)) is not separable. Hint:you might produce an isometry from `∞ (N) intoL∞ ([0,∞),m) and then use Exercise 5.2. find an uncountable set Λ ⊂ L∞ (R,m) as in the hypoth-esis of Exercise 5.1.

2. Use this result along with Exercise 5.10 in order to show L (Lp (([0,∞),B,m))) is not separable for all1 ≤ p <∞.

Exercise 5.12 (A “Martingale” Convergence Theorem). Suppose that Mn∞n=1 is an increasing se-quence of closed subspaces of a Hilbert space, H, Pn := PMn

, and xn∞n=1 is a sequence of elements fromH such that xn = Pnxn+1 for all n ∈ N. Show;

1. Pmxn = xm for all 1 ≤ m ≤ n <∞,2. (xn − xm) ⊥Mm for all n ≥ m,3. ‖xn‖ is increasing as n increases,4. if supn ‖xn‖ = limn→∞ ‖xn‖ < ∞, then x := limn→∞ xn exists in M and that xn = Pnx for all n ∈ N.

(Hint: show xn∞n=1 is a Cauchy sequence.)

Exercise 5.13. Let H be a Hilbert space. Use Theorem about orthonormal bases to show there exists a setX and a unitary map U : H → `2(X). Moreover, if H is separable and dim(H) = ∞, then X can be takento be N so that H is unitarily equivalent to `2 = `2(N).

Remark 5.3. Let H = `2 := L2 (N, counting measure),

Mn = (a (1) , . . . , a (n) , 0, 0, . . . ) : a (i) ∈ C for 1 ≤ i ≤ n ,

and xn (i) = 1i≤n, then xm = Pmxn for all n ≥ m while ‖xn‖2 = n ↑ ∞ as n → ∞. Thus, we can not dropthe assumption that supn ‖xn‖ <∞ in Exercise 5.12.

6


Exercise 6.1. Suppose (X,M, µ) and (Y,N , ν) are σ-finite measure spaces such that L2 (µ) and L2 (ν) areseparable. For f ∈ L2 (µ) and g ∈ L2 (ν) let (f ⊗ g) (x, y) := f (x) g (y) so that f⊗g ∈ L2 (µ⊗ ν) . If fn∞n=1

and gm∞m=1 are orthonormal bases for L2 (µ) and L2 (ν) respectively, show β := fn ⊗ gm : m,n ∈ N isan orthonormal basis for L2 (µ⊗ ν) . Hint: you might model your proof on the discrete analogue of thistheorem discussed in the notes.

6.1 Fourier Series Exercises

Exercise 6.2. Show that if f ∈ L1(

[−π, π]d)

and f (k) = 0 for all k then f = 0 a.e.

Exercise 6.3. Show∑∞k=1 k

−2 = π2/6, by taking f(x) = x on [−π, π] and computing ‖f‖22 directly and

then in terms of the Fourier Coefficients f of f.

Exercise 6.4 (Riemann Lebesgue Lemma for Fourier Series). Show for f ∈ L1(

[−π, π]d)

that f ∈

c0(Zd), i.e. f : Zd → C and limk→∞ f(k) = 0. Hint: If f ∈ L2(

[−π, π]d), this follows form Bessel’s

inequality. Now use a density argument.

Exercise 6.5 (Do not hand in). Suppose f ∈ L1([−π, π]d) is a function such that f ∈ `1(Zd) and set

g(x) :=∑k∈Zd

f(k)eik·x (pointwise).

1. Show g ∈ Cper(Rd).2. Show g(x) = f(x) for m – a.e. x in [−π, π]d. Hint: Show g(k) = f(k) and apply Exercise 6.2.3. Conclude that f ∈ L1([−π, π]d) ∩ L∞([−π, π]d) and in particular f ∈ Lp([−π, π]d) for all p ∈ [1,∞].

Notation 6.1 Given a multi-index α ∈ Zd+, let |α| = α1 + · · ·+ αd,

xα :=

d∏j=1

xαj

j , and ∂αx =

(∂

∂x

)α:=

d∏j=1

(∂

∂xj

)αj

.

Further for k ∈ N0, let f ∈ Ckper(Rd) iff f ∈ Ck(Rd)∩Cper(Rd), ∂αx f (x) exists and is continuous for |α| ≤ k.

Exercise 6.6 (Smoothness implies decay). Suppose m ∈ N0, α is a multi-index such that |α| ≤ 2m andf ∈ C2m

per(Rd)1.

1 We view Cper(Rd) as a subspace of H = L2(

[−π, π]d)

by identifying f ∈ Cper(Rd) with f |[−π,π]d ∈ H.


1. Using integration by parts, show (using Notation 6.1) that

(ik)αf(k) = 〈∂αf |ϕk〉 for all k ∈ Zd.

Note: This equality implies ∣∣∣f(k)∣∣∣ ≤ 1

kα‖∂αf‖H ≤

1

kα‖∂αf‖∞ .

2. Now let ∆f =∑di=1 ∂

2f/∂x2i , Working as in part 1) show

〈(1−∆)mf |ϕk〉 = (1 + |k|2)mf(k). (6.1)

Remark 6.2. Suppose that m is an even integer, α is a multi-index and f ∈ Cm+|α|per (Rd), then∑

k∈Zd

|kα|∣∣∣f(k)

∣∣∣2

=

∑k∈Zd

|〈∂αf |ek〉| (1 + |k|2)m/2(1 + |k|2)−m/2

2

=

∑k∈Zd

∣∣∣〈(1−∆)m/2∂αf |ek〉∣∣∣ (1 + |k|2)−m/2

2

≤∑k∈Zd

∣∣∣〈(1−∆)m/2∂αf |ek〉∣∣∣2 · ∑

k∈Zd

(1 + |k|2)−m

= Cm

∥∥∥(1−∆)m/2∂αf∥∥∥2H

where Cm :=∑k∈Zd(1 + |k|2)−m < ∞ iff m > d/2. So the smoother f is the faster f decays at infinity.

The next problem is the converse of this assertion and hence smoothness of f corresponds to decay of f atinfinity and visa-versa.

Exercise 6.7 (A Sobolev Imbedding Theorem). Suppose s ∈ R andck ∈ C : k ∈ Zd

are coefficients

such that ∑k∈Zd

|ck|2 (1 + |k|2)s <∞.

Show if s > d2 +m, the function f defined by

f(x) =∑k∈Zd

ckeik·x

is in Cmper(Rd). Hint: Work as in the above remark to show∑k∈Zd

|ck| |kα| <∞ for all |α| ≤ m.

Exercise 6.8 (Poisson Summation Formula). Let F ∈ L1(Rd),

E :=

x ∈ Rd :∑k∈Zd

|F (x+ 2πk)| =∞

and set

F (k) := (2π)−d/2

∫Rd

F (x)e−ik·xdx for k ∈ Zd.

Further assume F ∈ `1(Zd). [This can be achieved by assuming F is sufficiently differentiable with thederivatives being integrable like in Exercise 6.6.]

6.1 Fourier Series Exercises 21

1. Show m(E) = 0 and E + 2πk = E for all k ∈ Zd. Hint: Compute∫[−π,π]d

∑k∈Zd |F (x+ 2πk)| dx.

2. Let

f(x) :=

∑k∈Zd F (x+ 2πk) for x /∈ E

0 if x ∈ E.

Show f ∈ L1([−π, π]d) and f(k) = (2π)−d/2

F (k).3. Using item 2) and the assumptions on F, show

f(x) =∑k∈Zd

f(k)eik·x =∑k∈Zd

(2π)−d/2

F (k)eik·x for m – a.e. x,

i.e. ∑k∈Zd

F (x+ 2πk) = (2π)−d/2 ∑

k∈Zd

F (k)eik·x for m – a.e. x (6.2)

and form this conclude that f ∈ L1([−π, π]d) ∩ L∞([−π, π]d).Hint: see the hint for item 2. of Exercise 6.5.

4. Suppose we now assume that F ∈ C(Rd) and F satisfies |F (x)| ≤ C(1 + |x|)−s for some s > d andC <∞. Under these added assumptions on F, show Eq. (6.2) holds for all x ∈ Rd and in particular∑

k∈Zd

F (2πk) = (2π)−d/2 ∑

k∈Zd

F (k).

For notational simplicity, in the remaining problems we will assume that d = 1.

Exercise 6.9. Let µ be a finite measure on BRd , then D := spaneiλ·x : λ ∈ Rd is a dense subspace of Lp(µ)for all 1 ≤ p <∞. Hints: from class and the notes we know that Cc(Rd) is a dense subspace of Lp(µ). Forf ∈ Cc(Rd) and N ∈ N, let

fN (x) :=∑n∈Zd

f(x+ 2πNn).

Show fN ∈ BC(Rd) and x→ fN (Nx) is 2π – periodic and so x→ fN (Nx) can be approximated uniformlyby trigonometric polynomials. Use this fact to conclude that fN ∈ DLp(µ). After this show fN → f in Lp(µ).

Exercise 6.10. Suppose that µ and ν are two finite measures on Rd such that∫Rd

eiλ·xdµ(x) =

∫Rd

eiλ·xdν(x) (6.3)

for all λ ∈ Rd. Show µ = ν.Hint: Perhaps the easiest way to do this is to use Exercise 6.9 with the measure µ being replaced by µ+ν.

Alternatively, use the method of proof of Exercise 6.9 to show Eq. (6.3) implies∫Rd fdµ(x) =

∫Rd fdν(x) for

all f ∈ Cc(Rd) and then apply Corollary ??.

Exercise 6.11. Again let µ be a finite measure on BRd . Further assume that CM :=∫Rd e

M |x|dµ(x) <∞ for

all M ∈ (0,∞). Let P(Rd) be the space of polynomials, ρ(x) =∑|α|≤N ραx

α with ρα ∈ C, on Rd. (Notice

that |ρ(x)|p ≤ CeM |x| for some constant C = C(ρ, p,M), so that P(Rd) ⊂ Lp(µ) for all 1 ≤ p < ∞.) ShowP(Rd) is dense in Lp(µ) for all 1 ≤ p <∞. Here is a possible outline.

Outline: Fix a λ ∈ Rd and let fn(x) = (λ · x)n/n! for all n ∈ N.

1. Use calculus to verify supt≥0 tαe−Mt = (α/M)

αe−α for all α ≥ 0 where (0/M)

0:= 1. Use this estimate

along with the identity

|λ · x|pn ≤ |λ|pn |x|pn =(|x|pn e−M |x|

)|λ|pn eM |x|

to find an estimate on ‖fn‖p .2. Use your estimate on ‖fn‖p to show

∑∞n=0 ‖fn‖p <∞ and conclude

limN→∞

∥∥∥∥∥eiλ·(·) −N∑n=0

infn

∥∥∥∥∥p

= 0.

3. Now finish by appealing to Exercise 6.9.


6.2 Extra Problems to ponder (Do not hand in.)

Exercise 6.12 (Heat Equation 1.). Let (t, x) ∈ [0,∞)× R → u(t, x) be a continuous function such thatu(t, ·) ∈ Cper(R) for all t ≥ 0, u := ut, ux, and uxx exists and are continuous when t > 0. Further assumethat u satisfies the heat equation u = 1

2uxx. Let u(t, k) := 〈u(t, ·)|ϕk〉 for k ∈ Z. Show for t > 0 and k ∈ Zthat u(t, k) is differentiable in t and d

dt u(t, k) = −k2u(t, k)/2. Use this result to show for t > 0 that

u(t, x) =∑k∈Z

e−t2k

2

f(k)eikx (6.4)

where f(x) := u(0, x) and as above

f(k) = 〈f |ϕk〉 =

∫ π

−πf(y)e−ikydy =

1

2π

∫ π

−πf(y)e−ikydm (y) .

Notice from Eq. (6.4) that (t, x)→ u(t, x) is C∞ for t > 0.

Exercise 6.13 (Heat Equation 2.). Let qt(x) := 12π

∑k∈Z e

− t2k

2

eikx. Show that Eq. (6.4) may be rewrittenas

u(t, x) =

∫ π

−πqt(x− y)f(y)dy

andqt(x) =

∑k∈Z

pt(x+ k2π)

where pt(x) := 1√2πt

e−12tx

2

. Also show u(t, x) may be written as

u(t, x) = pt ∗ f(x) :=

∫Rd

pt(x− y)f(y)dy.

Hint: To show qt(x) =∑k∈Z pt(x+k2π), use the Poisson summation formula (Exercise 6.8) and the Gaussian

integration identity,

pt(ω) =1√2π

∫Rpt(x)eiωxdx =

1√2πe−

t2ω

2

. (6.5)

Equation (6.5) will be discussed in more detail later in the class.

Exercise 6.14 (Wave Equation). Let u ∈ C2(R × R) be such that u(t, ·) ∈ Cper(R) for all t ∈ R.Further assume that u solves the wave equation, utt = uxx. Let f(x) := u(0, x) and g(x) = u(0, x). Show

u(t, k) := 〈u(t, ·), ϕk〉 for k ∈ Z is twice continuously differentiable in t and d2

dt2 u(t, k) = −k2u(t, k). Use thisresult to show

u(t, x) =∑k∈Z

(f(k) cos(kt) + g(k)

sin kt

k

)eikx (6.6)

with the sum converging absolutely. Also show that u(t, x) may be written as

u(t, x) =1

2[f(x+ t) + f(x− t)] +

1

2

∫ t

−tg(x+ τ)dτ. (6.7)

Hint: To show Eq. (6.6) implies (6.7) use

cos kt =eikt + e−ikt

2,

sin kt =eikt − e−ikt

2i, and

eik(x+t) − eik(x−t)

ik=

∫ t

−teik(x+τ)dτ.

6.2 Extra Problems to ponder (Do not hand in.) 23

Exercise 6.15. Suppose H is a Hilbert space and Hn : n ∈ N are closed subspaces of H such that Hn ⊥Hm for all m 6= n and if f ∈ H with f ⊥ Hn for all n ∈ N, then f = 0. For f ∈ ⊕∞n=1Hn, show the sum∑∞n=1 f (n) is convergent in H and the map U : ⊕∞n=1Hn → H defined by Uf :=

∑∞n=1 f (n) is unitary.

Exercise 6.16. Suppose (X,M, µ) is a measure space and X =∐∞n=1Xn with Xn ∈M and µ (Xn) > 0 for

all n. Then U : L2 (X,µ)→ ⊕∞n=1L2 (Xn, µ) defined by (Uf) (n) := f1Xn

is unitary.

Exercise 6.17 (Knowledge of analytic functions is helpful). Again let µ be a finite measure on BRd

but now assume there exists an ε > 0 such that C :=∫Rd e

ε|x|dµ(x) <∞. Also let q > 1 and h ∈ Lq(µ) be a

function such that∫Rd h(x)xαdµ(x) = 0 for all α ∈ Nd0. (As mentioned in Exercise 6.17, P(Rd) ⊂ Lp(µ) for

all 1 ≤ p <∞, so x→ h(x)xα is in L1(µ).) Show h(x) = 0 for µ– a.e. x using the following outline.Outline: Fix a λ ∈ Rd, let fn(x) = (λ · x)

n/n! for all n ∈ N, and let p = q/(q − 1) be the conjugate

exponent to q.

1. Use calculus to verify supt≥0 tαe−εt = (α/ε)

αe−α for all α ≥ 0 where (0/ε)

0:= 1. Use this estimate

along with the identity

|λ · x|pn ≤ |λ|pn |x|pn =(|x|pn e−ε|x|

)|λ|pn eε|x|

to find an estimate on ‖fn‖p .2. Use your estimate on ‖fn‖p to show there exists δ > 0 such that

∑∞n=0 ‖fn‖p < ∞ when |λ| ≤ δ and

conclude for |λ| ≤ δ that eiλ·x = Lp(µ)-∑∞n=0 i

nfn(x). Conclude from this that∫Rd

h(x)eiλ·xdµ(x) = 0 when |λ| ≤ δ.

3. Let λ ∈ Rd (|λ| not necessarily small) and set g(t) :=∫Rd e

itλ·xh(x)dµ(x) for t ∈ R. Show g ∈ C∞(R)and

g(n)(t) =

∫Rd

(iλ · x)neitλ·xh(x)dµ(x) for all n ∈ N.

4. Let T = supτ ≥ 0 : g|[0,τ ] ≡ 0. By Step 2., T ≥ δ. If T <∞, then

0 = g(n)(T ) =

∫Rd

(iλ · x)neiTλ·xh(x)dµ(x) for all n ∈ N.

Use Step 3. with h replaced by eiTλ·xh(x) to conclude

g(T + t) =

∫Rd

ei(T+t)λ·xh(x)dµ(x) = 0 for all t ≤ δ/ |λ| .

This violates the definition of T and therefore T =∞ and in particular we may take T = 1 to learn∫Rd

h(x)eiλ·xdµ(x) = 0 for all λ ∈ Rd.

5. Use Exercise 6.9 to conclude that ∫Rd

h(x)g(x)dµ(x) = 0

for all g ∈ Lp(µ). Now choose g judiciously to finish the proof.

Documents

Part I 240B Homework Exercises (Winter 2012)bdriver/240b-c_2012/Homeworks/240B_Home… · 10 3 HomeWork #3 Due Friday January 27, 2012 1.If fx ngis Cauchy sequence, then there is