Texts: Elementary
Classical Analysis (required),
by Marsden and Hoffman, 2nd Edition; Freeman Press
Proofs are at the end of chapters and must
be read.
The Way of Analysis (recommended),
by Strichartz; Jones and Bartlett Press
Useful for learning how to prove.
Notes: Supplementary
Handouts will be used especially in the
beginning of the course to teach proving techniques.
Prerequisites: Multivariable
Calculus and Linear Algebra are
required. This course is significantly easier to handle if you
have had experience proving theorems. It is recommended that
you review proofs you have studied in the past including epsilon
delta proofs from Calculus.
Grading Policy:
Homework,
40%: The problems in this course are difficult
and
cannot be completely tested on exams. Thus the homework is
a significant portion of your grade. Sometimes you will be
required to resubmit an assignment to improve your grade.
No late assignments will be accepted without a doctor's
note. Assignments are due at the beginning of class if
due
Thursday and at 5pm if due Tuesday. You may consult with
classmates but be sure to do most of the work yourself.
Starting after week 4, homework due on Thursdays
will be short
assignments on old material which will be graded in their entirety.
Homeworks due on Tuesdays will be as before.
Exam
I, 15%: This exam was scheduled after
the fourth week.
It consisted of four problems, the first three on a Monday evening
with no time limit (~2 hours) and the fourth on a Thursday with a 20 min
time limit. Problem 1 was a choice of one of two proofs by
contradiction (20%). Problem 2 was a choice of one of two limit proofs
(20%).
Problem 3 was a choice of two four part problems one on intersections
the other on equivalence classes (40%). Problem 4 is a harder proof
by
contradiction (20%) see the email.
Exam II, 15%:
This exam will cover the topology of metric spaces:
continuity, open sets, closed sets, boundaries, compactness,
connectedness etc. It will be given one week after this material
is
covered.
Final
30%: This exam will be given during
the finals week.
Syllabus: (This syllabus
will be updated as the course progresses)
Week 1:
Introduction to Analysis, What is Proof?, Handout,
Contradictions, Mappings, Equivalence Classes, Handout
HW: Handout
1-3, all problems, due Tuesday Sept 8
Handout 4-5, all problems, due Thursday Sept 10
Our new classroom is 225 NEB behind Shaffer Hall.
If you did not get email about this, send me an
email!
Week 2:
Denumerability and the Rationals, Handout, 1.1
Sequences in the Rationals, Handout, 1.2
HW: Handout 6-7, all problems,
due Tuesday Sept 15
Handout 8, all problems, 1.1/5 due Thursday Sept 17
Week 3:
Construction of the Real Line and Completeness
Unique Decimal Expansions, [0,1] is not denumerable
HW: Handout 9-10, due Tuesday
Sept 22
1.2/1, 2, 4, 5 due Thursday Sept 24
Week 4:
Euclidean space, Cauchy Schwartz, Complex , 1.6,1.8
Metric Spaces 1.7,
HW: Due Thursday Oct 1 but good
review for the exam:
Handout 11 (on my door), all problems
(Read all of1.4) 1.4/ 1, 4, 5 (use negation for 5)
(Read 1.8 only pp70-72), verify that the complex plane
is a field, 1.8/1, 5
After the exam, read 1.7
Week 5:
Exam I Monday
Sept 29 6-8 pm or 7-9pm,
Krieger hall Room 300, Problems 1-3
Thursday (20 min in class Exam I Problem 4)
If you did not get the email, send a note.
Open sets and Interiors, 2.1
Interiors and Closed Sets, 2.2-2.3
HW due Tue: 2.1/ 1, 3, 4 (draw B if C=[0,1]x{0}) , 2.2/3,
4, 2.3/ 4
HW due Th : 1.2/ 4 give
explicit counter example or prove with explicit N
depending on epsilon, p97/ 3a,b note that ]0,1[=(0,1), p97/6
Week 6:
Closure, Boundaries, infs and sups 2.4-2.6, 1.3
Sequences, Cluster Points, limsups, 2.7-2.8, 1.5
HW due Wed: (Will grade *'ed only)
2.7/2*,3*, 2.4/1,2*,3, 6*,
2.5/1, 2*, 1.3/2*, 97/1
HW due Friday: 2.2/3, 2.3/1,2*,6*,1.4/3,
98/10, 99/15
Week 7:
Sequencial Compactness 3.1.1
Compact Sets and Bolzano Weierstrass 3.1
HW due Tuesday: 1.5/ 1,2*,4*(prove or provide a counterexample)
2.8/2*,3,4*, 3.1/1,2*,4*,5
HW due Thursday:
p143-5/1,2*(no proof), 4*, 7, 8*,9a, 13, 15a*,18*
Week 8:
Bolzano Weierstrass Continued
Euclidean Space and Heine Borel 3.2 (Handout),
Hilbert Space (Handout)
HW due Wednesday: 3.2/1 , 4*, Heine-Borel
Handout/1*, 3*,4*,
Hilbert Space Handout/ 1*, 2*, 4*,5, 6*, 7*
Show that if Vi is a sequence in Hilbert space that
converges to V, then each component converges
to the appropriate component of V. Give an example
showing that even if each component converges, the
sequence itself may not converge.
Extra Credit from Heine Borel Handout due Friday
Hilbert Space/ 3 is also extra credit but no deadline yet.
Week 9:
Continuity 4.1 (not Thm 4.1.4 iii and iv),
Path connected and Connected Sets 3.4-5
HW due Tuesday: 4.1/1*,
5, p231/1, problems 1*, 2* and 3* below
3.4/1(no proofs), 3.5/1,2*
1) Let g: A to R where A=R\{0}. Let g(x)= (cos(x)-1)/x.
Prove that lim
g(x) = 0 using the defn.
x to 0
2) Let F:R to R^2 be defined as F(t)=(x(t), y(t)) where x and y are functions
from R to R. Prove that F is continuous at s iff x and y are continuous
at s.
3) Let g: (-pi/2, pi/2) to R be defined as g(x)=tan(x). a)
Prove that
g is continuous at any point in (-pi/2, pi/2). b) prove that
pi/2 is
an accumulation point of (-pi/2, pi/2). c) prove that there
does not
exist L such that
lim g(x)
= L .
x to pi/2
HW due Thursday: p172-6/1,
4*, 5b*, 12*, 14*, 17, 18*, 33, 37,
Week 10:
Continuous Images and Preimages of sets 4.1-4.2
Intermediate Value Theorem 4.4-4.5 (purely Real)
Exam II Monday Nov 9: All
material weeks 4-9 and 4.2.
2 hours (come between 6-9pm) Maryland Hall 114.
No Homework this week,
Week 11: Uniform
Continuity 4.6, Differentiation of functions on the Real line 4.7
Integration, Fundamental Theorem of Calculus, 4.8, 8.1
HW due Wednesday: Differentiation:
4.6/1, 2*, 3, 6*, 4.7/2*, and 1* below
Integration: p234-6/22*, 44*, 45 8.1/2*,3, 6* and
2* below
Continuity Review: 3* below.
1) Let f(x) and g(x) be differentiable on R, f(y)=g(y)=0 and g'(y)
not 0.
Use the mean value theorem and definition of limit to prove
L'hopital's Rule: limit as x approaches y of f(x)/g(x)
= f'(y)/g'(y).
2) Let F(x)=0 when x is irrational and F(x)=1 when x is rational.
Prove that F is not Riemann Integrable (that is, show that
inf upper sums > sup lower sums). Hint: Remember that
between any pair of real numbers, x<y, there is a rational
number q and an irrational r between them.
3) Let f: R to R be continuous everywhere. The support
of
f is the set A = Cl({ x : f(x) does not equal 0}). Prove that
if f has compact support then f is bounded.
Week 12:
The Space of Continuous Functions, Uniform Convergence 5.1, 5.5
Integration and differentiation of limits, Function Spaces
(need only know: 5.2.1, 5.3.1, 5.3.3, 5.3.5, 5.3.6, from 5.2-3).
HW due Tuesday Nov 24:
5.1/1, 2*,3, 5.2/1, 5.3/1*, 2*, 3*
Function Spaces Handout (use defn of support as above)/ 1*, 2**, 3*, 4*,5*
HW due Friday Dec 4: p174-6/
21*, 39*,
p234-6/ 26*, 38*, 39, 42* , 45, p316/ 2*, 10 (only sequences from
2)*
Review Exam II including problems you didn't choose.
Week 13:
Volume, Measure Zero and Denumerability 8.2
Review
No more HW due. Please fill out the Teaching
Evaluation before the final.
Review Problems: 5.1/4, p231-4/1a, 3, 7, 11a (just do f is cont implies
graph is path conn),
15, 19, 21, 29, 33 p316-318/ 1, 5, 13, 19 (show the sum conv unif
on [-r,r] for any r),
29ac, 33, 37, ...
Final:
Friday December 11, Shaffer Hall Room 1, 2-5pm
There will be five parts:
I. Short Answers (these will be longer than the ones in the second
exam but not as many) negating statements, verifying something
is a metric, find limsup/inf with no justification, identify compact
open, closed and connected sets, and boundaries in Euclidean space.
II. (Chose between two explicit sequences of functions),
Prove or prove not uniformly conv, prove if the derivatives
conv and/or the integrals. (be sure to know theorems of week 12
and techniques I went over last Thursday)
III. (chose one of two) Proofs regarding uniform continuity,
Riemann Integrability and differentiability. (week 11)
(Know these definitions)
IV. (chose one of two) Proofs about connectedness and
continuity in metric spaces (week 9-10)
V. (chose one of two) Proofs about the metric space properties of sets
in
C([0,1]) (these will include new concepts) Be sure to know the
definitions of closed, open, compact, boundary, connected, closure,
interior so that you can apply them here as well as in I. Write down
the definitions before you apply them. (weeks 5-7,12)