##### Analysis II
110.406,  ThFr 10:30-11:50, Krieger Hall 304.

Professor:  Christina Sormani
Kriegar 220,  516-6637                Office Hours: Tuesdays 5-6pm
sormani@math.jhu.edu                                 more by appointment

Texts:  Elementary Classical Analysis
by Marsden and Hoffman, 2nd Edition; Freeman Press
Proofs are at the end of chapters and must be read.

The Way of Analysis,
by Strichartz; Jones and Bartlett Press
We will be refering to this when we study Lebesgue Measure.
See below for specific sections we will be using.

Prerequisites:   Multivariable Calculus, Linear Algebra,
and Analysis I are required.    A detailed syllabus of
material covered in Analysis I is available here.

Homework, 40%:   The problems in this course are difficult and
cannot be completely tested on exams.  Thus the homework is
accepted without a doctor's note.    Assignments are due at the
beginning of class on Thursday.    You may consult with
classmates but be sure to do most of the work yourself.   Only
starred problems will be graded but all problems are useful for
understanding and practice.  Try to do the reading and the
problems in the order they are assigned.

Exam I,  15%Lebesgue Measure and Integration given after the fifth
week of class.

Exam II, 15%:  Hilbert and Banach Spaces,  Fourier Series,  Stone
Weierstrass Thm,  Arzela Ascoli Thm,  given after the tenth week of class

Final 30%:   All Previous material and:  Contraction Mapping Thm,
Diffeomorphisms, Inverse and Implicit Function Theorems, and more
if time permits.  This exam will be given during the finals week.

Schedule and Homework:
(This syllabus will be updated as the course progresses)

Week 1:    Review of Riemann Integration and convergence of functions  Marsden 5.1-5.5
Review of Compactness, Heine-Borel Thm  Marsden  3.1-3.2
Length, Countable Additivity, Sigma Fields of Sets  Strich + Handout I: 14.1.1-2

HW1:  Handout I (14.1.1-2) do problems and reading in order:
14.1.1/1**, 14.1.2/1**,2*,3**,4**,5*  (review topics listed above)

Week 2:  Sigma Fields, Borel Sets, Pointwise Limits of Functions  Strich + Hand I:14.1.3
Lebesgue Measure and Open Sets in R ,  Strichartz + Handout I:14.1.4

HW2:  Handout (14.1.3-4) do problems and reading in order:
14.1.3/1,2,3*,4*,5*,6*,7,8,9*; 14.1.4/1*,2**,3** (review connected sets)

Week 3:  Lebesgue Measure and the Splitting Condition  Strich + Hand I: 14.1.5
Caratheodory Theorem  Hand I: 14.2 (Strichartz page 645-6)

HW3:  Handout (14.1.5-14.2) do problems and reading in order:
(you might want to quickly review 14.1.1-4 to recall what has been proven)
14.1.5/ 1** (sigma subadditivity), 2 (monotonicity),
14.2.1/ 1**, 2**, 3, 4**;   14.1.7/7*, 8*,

Week 4:  Monotone Convergence Theorem Strichartz 14.3.2
Fatou's Theorem and Integrable Functions  Strich 14.3.3

HW4: 1)* Prove the theorem that if f is a function  (values in 0 to infinity)
such that for any real number, a, f^{-1}((a, infinity)) is measurable,
then f is a measurable function as in defn on p 656.  (use defn
of Borel sets and sigma fields and the theorem which says the
the Borel sets are generated by intervals).
2)* Let P ={0 < y1< y2,...< yn < infinity} and Q={0 < z1< z2,...< zn < infinity} be
partitions.  Q is a subpartition of P if  {0, y1,y2,...yn, infinity} is a
subset of {0, z1, z2,...zn, infinity}. Prove that L(f,P) <= L(f,Q) if
Q is a subpartition of P (Hint: rename the z's using the y's).
3)* Prove that     f(x)=sup{fn(x), n in N}     iff     for all a in R
f^{-1}((a, infinity)) = countable union of fn^{-1}((a, infinity))
4) * Write up a proof of Theorem 14.3.3 b and c
5) * Write up a detailed proof of Theorem 14.3.4
6) ** Use the monotone convergence theorem (14.3.2) and well chosen
subpartitions to prove that the defn of Lebesgue Integral using
limit of the special sequence of partitions Pn (p655) is equal
to the supremum of L(f,Q) over all partitions Q of [0, infinity].
Remember to use the definition of supremum.
Exercises: 14.1.7/ 15*;  14.3.5 /2*, 3*, 17.
Extra Credit:  14.3.5/7 (if you use the hint, prove the hint). (due March 11)

Week 5:  Almost Everywhere,    Strich 14.3.4
Review of Equivalence Classes and the Real Line (see handout from Analysis I)
The Lebesgue Function Space,  L1 ,   Strichartz 14.4.1

HW 5:  due Friday March 5 (but best if done by Tuesday so you can ask questions and
use as review for the exam after reading 14.3.4) 14.3.5/ 4*,8*, 9, 10*, 11*, 14**
14.1.7/1,2, 14*,   (read 14.4.1 up to Thm 14.4.1 and do the following:)
1)* Prove that "f=g almost everywhere" is an equivalence relation on the
space of measurable functions, and that the space of measurable
functions modulo this equivalence, L1, has a well defined Lebesgue integral.
2)* Prove that the characteristic function of the rationals is equivalent to
the zero function.
3)* Use 14.3.5/ 14 to justify that in  L1, the || f-g ||1 is a metric, where f and g
are really equivalence classes of functions.

Exam I:  Lebesgue Measure and Integration (Weeks 1-4)
Thursday, March 4, 4-8pm.  Choose a 3 hour subset of this time slot.
Be sure to know the following defns and thms and be able to use them:
Structure Theorem of Open sets, ptwise convergence of funtions,
Field, sigma field, Borel sets, measure (p634), monotonicity of measures,
continuity from below, conditional continuity from above, subadditivity,
formula for Lebesgue measure, defn of inf and sup with epsilons,
measurable sets and the splitting condition, defn of Lebesgue Integral
using special partitions (p655), thm from HW4/6, Leb Int of a simple function,
characteristic functions, simple functions, measurable functions, the thm
in italics on the bottom of p659, examples 1&2 (p660), Monotone Conv Thm,
Thm 14.3.3, Fatou's Thm, Integrable, Examples 1&2 from week 5 lesson 1,
extension of 14.3.3 to integrable functions, Dominated Convergence Thm.
Be sure to be able to do proofs: verifying that something is a measure,
verifying that something is a field/sigma field,  involving converting
info about functions into info about sets (like in Lemma 14.1.3 and HW4/3),
computing the Lebesgue measure of a set using the formula and/or theorems,
verifying that a set is measurable using the splitting condition and the formula
for Lebesgue Measure, computing the Lebesgue Integral of a function
using theorems and/or definition, verifying that a function is measurable.

Week 6:   Completeness of LStrichartz 14.4.1, Thm 14.4.1 to end.
The Hilbert Space, L2 ,  Strichartz 14.4.2, Marsden 10.1 -2
Review of Linear Algebra, Marsden 1.7, (also Strichartz 9.1.2-3)

HW6: 14.4.4/ 1**,3**,4**,10**, 9.1.4/1*,2*
1) Prove that if f and g are measurable then fg is measurable.

Week 7:  Completeness of L2 ,   Lvs L2   Strichartz 14.4.2,
Orthogonal Families of Functions,  Marsden 10.1-2,
Density of Continuous Functions, Strichartz Thm 14.4.5

(1)* If f is in L2([0,2pi]), is f in L1([0,2pi])?  justify.
(2)* If fn converges to f in L2([0,2pi]), does it converge in L1([0,2pi])?  justify.
(3)** Prove that fn(x)=sum {j=1 to n} of cos(jx)/j does not converge pointwise
but is Cauchy in L2([0, 2pi]) so it converges in L2([0,2pi]).
(4)*  suppose fn converges to f in L2([0,2pi]), and g is in L2([0,2pi]),
show that the real numbers <fn,g> converge to <f,g>.
Recall "convergence in the mean" is L2 convergence.
10.1/5*, 10.2/4a*, b*  (Treat all integrals in Marsden as if they were Lebesgue Integrals)
For the next two problems, we use the definition: "f is in the span of a system of
functions, PHI0, PHI1, PHI2,..." iff there exists a1,a2,a3... such that
f is the L2 limit of the partial sums, SUM{j=1 to n} aj PHIj.
(5) prove that thm 10.2.4 can be stated as :  f is in the span of an orthonormal
system  PHI0, PHI1, PHI2...   iff    ||f||^2= sum (<f, PHIj>)^2.
(6)*  Suppose fn are a sequence of functions which are in the span of
an orthonormal system of functions PHI0, PHI1, PHI2..., and suppose
fn converges to f in L2([0,2pi]), prove that f is in the span as well using
thm 10.2.4 and problem 4 above.
Read Strichartz, thm 14.4.5 (and proof)

Spring Break

Week 8: Stone Weierstrass and Fourier Series,  Marsden 5.8, 10.3 (Strich 14.4.3)
Marsden 10.3 lemma 2 p624 replace with Strichartz 14.4.5
Gibb's Phenomenon, Marsden Thm 10.5.2, Example 10.5.4,

HW8:  Read Marsden 5.8 and proofs of all theorems from that section.
1)Let  nC = k! (n-k)! / n! and define qn :[-M,M] to R as in class:
qn(t) = Sum (k=1 to n) nC(t/(2M) + 1/2)^k (1-( t /(2M) + 1/2) )^(n-k)  | 2Mk/n-M |
Prove that  qn converges to |t| uniformly on [-M,M].  (Imitate proof of Thm 5.8.1)
5.8/1,2*,3,5*, 5.8/6 (contrast with Fourier Series)*,
Read Marsden 10.3 and proofs of all theorems in this section except 10.3.2,
and replace lemma 2 p624 with Strichartz Thm 14.4.5 from before break.
10.3/ 1, 3, 4*,5 a b* c d* e*
Read Marsden 10.5.2 (and its proof) and example 10.5.4
10.5/ 1a (graph n=5, n=10 and n=20 and f itself using a computer, be sure the
Gibb's Phenomenon appears (may need to increase number of x plot points))**,

Week 9: C([0,1]) and Equicontinuity,  Marsden 5.6
Arzela Ascoli Theorem,  Marsden 5.6
If time, Uniform Conv Thm of Fourier Series, Marsden 10.6.1

HW9: 10.5/5* , Read Theorem 10.6.1 and proof, 10.6/5*,
Review Marsden 5.5, 5.5/4*  ,5* ,
Review Marsden Thms 4.2.2 and 4.4.1, Read Marsden 5.6 and proofs,
5.6/1* ,2(if no give a counter example, if yes prove it)**  , 3a**  , 4*

Week 10:  Peano Existence Theorem  Strich 11.2.2
Contraction Mapping Principle Marsden 5.7

Exam II:  Function Spaces L1, L2,  and C([0,1]),  (Weeks 5-9).
Thursday April 15, 4-8pm (choose 3 hours), Room 211 Krieger Hall
Review subadditivity, continuity from below, cond contin from above
Review relationship between L1 and Lebesgue measure (as in the proof
of S14.3.6 and of the completeness of L1).
Think about L1 vs L2 vs C([0,1]) vs ptwise convergence of functions.
Know the proof of why Fourier series converge in the mean,
completeness of  orthonormal basis, projections (M10.2.5)
Know how to apply and state Arzela Ascoli Thm, Stone Weierstrass thm,
Cauchy-Schwartz (S14.4.2) , density of continuous functions (S14.4.5),
integral convergence theorems.
The exam will have many short problems.  You will be required to do
6 of the 7 starred problems and 4 more problems.  Whether or not
you do a problem, you should feel free to use it to prove a subsequent
problem.   Some problems will be very difficult if you do not take
advantage of previous problems.  So prepare to read the exam in order.
There will be extra office hours Tuesday if anyone wishes.  Be sure to
understand the proofs mentioned above.

Week 11:  Uniqueness of solutions to Ordinary Differential Equations  Marsden 5.7
Riemann Integration to Lebesgue Integration
No Homework this week but may wish to start HW 10 problems from 5.7 and 316-319.

Week 12:  Higher Dimensional Derivatives, Diffeomorphisms  Marsden 6.1-6 (a lot to read!)
Inverse Function Theorem Marsden 7.1

HW 10:   Read all of 5.7 inc proofs, 5.7/2*, 3, 4*, 5, 6*, 8*,
p316-319/14*, 26**,

Week 13:  Implicit Function Theorem  Marsden 7.2-3
Review for Final April 30 10:30-12:00.

HW 11:   (due Tuesday May 4 during office hours)
Review Marsden Chap 5, p321/ 43*, 45*, 46*
1)* Let x=rcos(t) y=r sin(t).  Let  F(r,t)=(x(r,t), y(r,t)).
Approximate F near (1,0) with a linear function G(r,t).
Draw the image of the r-t grid under F and under G and compare.
2)* Find U and V such that F is a diffeomerphism from U to V and graph
the sets.  Discuss what goes wrong if U contains (0,0) by drawing
the image of the grid under F and the image of the grid under its
linear approximation H near (0,0).
7.1/1, 2*, 3*, 4*, 5;
Read 7.2-7.3 inc proofs  7.2/1, 2*, 3, 4*, 5

Final Exam:   Friday May 7, 2-5 pm
All material (with some emphasis on Weeks 10-13)