Math 414 and Math 514 (Phillips):
Introduction to Analysis 2
This is the home page for N. C. Phillips' Introduction to Analysis 2
(Math 414 and Math 514) the University of Oregon, Winter quarter 2019.
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Basic information

Course number: Math 414,
CRN 24367,
and Math 514,
CRN 24376,

Course title: Introduction to Analysis 2.

Time and place: MWF 11:0011:50 am, 303 Deady.
Discussion section: Tu 1111:50 am, room to be announced.
I expect be away on a research visit during the 7th week of classes,
maybe into the 8th week.
The lectures will probably be given by substitutes.

Instructor: N. Christopher
Phillips.

Office: 105 Deady.
The office is on the main level at the east end of the building,
opposite the stairs
(very close to the classroom).
Please knock.
I can't leave my door open, because if I do I get too many people
asking to borrow my telephone or pencil sharpener, or
where to find the math department office
or nonexistent rooms (such as 350 Deady).

Office hours: MF 2:303:20 pm, Th 1111:50 am,
or by
appointment.

Email.
The subject line of your message should start with
"M414" (without the quotation marks),
followed by your last name,
then first initial.
I will use email occasionally to distribute general announcements.
I will give short replies to emailed questions.
I don't type at a reasonable
speed, so I will rarely answer complicated questions by email.
Please come to office hours instead.
When emailing me, please use plain text
(7 bit ASCII)
only.
That is, only the characters found on a standard English
language keyboard; no curved quotation marks, curved apostrophes,
accented letters, Greek letters, etc.
Use TeX pseudocode for mathematical symbols,
including Greek letters.
In particular:

No html encoded (web page format, or "styled") messages.
See "Configuring
Mail Clients to Send Plain ASCII Text"
for how to turn off html.
(The University of Oregon webmail program is apparently capable
of sending plain text email.)

No binary files or attachments (except by prior arrangement).

No Microsoft Word files.
I do not accept these under any circumstances,
since I don't have software that reads them.

No mime encoding or other encoding of ordinary text messages.

Course description:
The three quarter sequence Math 413[513]Math 415[515]
give a rigorous presentation of calculus of one and several
variables
(the material of Math 251Math 253 and Math 281Math 282).
In Math 414[514],
we will cover differentiation, Riemann integration,
and continuity, differentiation, and integration of
sequences and series of functions.
We will also give rigorous constructions
of the standard functions from elementary calculus courses,
and possibly some others
(such as the Gamma function).
We will look briefly at Fourier series.

Textbook:
W. Rudin, Principles of Mathematical Analysis
3rd edition.

Syllabus (pdf);
LaTeX.
Everything here is also there,
with better formatting for the mathematical expressions
and sometimes in more detail,
but without links.
Exception: My email address isn't there,
since spammer address harvestors read pdf files.

Notes on Terminology:
The book contains a few oddities of notation and terminology which
I will not follow.
Open intervals are referred to as ``segments''.
The sets of real numbers
and rational numbers are called R and Q
instead of the
more usual blackboard bold (or boldface) versions.
(In particular, I will feel free to use
the letters R and Q for
other purposes.)
In contrast to this, the book suddenly switches from R to
R
when the function is supposed to take values in R^n
instead of a general metric space;
I will make no such distinction.
Countable sets are required to be infinite
(but some of the writing is inconsistent with this convention);
I will consider finite sets to be countable.
See the corresponding part of the
pdf
version of the syllabus;
for more,
on the set of natural numbers,
on "increasing" vs. "strictly increasing"
vs. "nondecreasing",
etc.
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Grading
Homework
Contribution to final grade: approximately 20%.
Due every Monday, except as otherwise stated.
The homework will be hard, some of it quite hard, but it is an
essential part of the course.
The homework consists of proofs.
This means that everything must be proved.
In particular, if a problem asks for an example or counterexample,
you must prove that your example has the required properties.
If a problem asks whether something is true, you must not only decide
whether it is, but also provide a proof or counterexample as
appropriate; just as above, counterexamples must be proved to
have the required properties.
Proofs should be clear, correct, and complete.
They should use complete sentences and correct punctuation.
They should have all quantifiers in the right places.
They should be in logical order.
For example,
for a proof of convergence of a sequence from the definition,
start with "Let epsilon be greater than zero"
and end with the statement that
something is less than epsilon.
If there are subsidiary arguments in a proof,
say what is being claimed,
say that you are proving it,
prove it,
and say when the proof of your claim is done.
A reader who can't follow any of the mathematics should
nevertheless be able to easily tell
where the proof of your subsidiary claim starts
and where it ends.
I expect to post on the course website
at least sketches of some solutions after the
homework is turned in.
Some procedural points:

Staple pages together.
Don't fold or tear corners or fold in half.

If you cooperate with someone else on any particular problem,
you must acknowledge the cooperation (including the name of the person)
in your solution.
One Midterm
Contribution to final grade: approximately 30%.
Date: Monday 11 February (week 6).
The midterm from the last time I taught the course is
here
(pdf);
LaTeX.
Its solutions are
here
(pdf);
LaTeX.
The format of the midterm this time may be a bit different.
A large portion of the midterm will ask for proofs which you will
have to devise for yourself during the exam period.
This midterm was given during a two hour period outside of class.
I am willing to allow people to start an hour early
or continue an hour late,
or to arrange to give the entire exam during a longer period outside
of the normal class time.
(This was done the last time I taught the course.)
Such a procedure can only be followed if suitable rooms and times
can be found, and if everybody enrolled in the class agrees to it.
Final Examination
Contribution to final grade: approximately 50%.
Date: Monday 18 March, 10:15 am12:15 pm, in our usual classroom.
(The final exam schedule is
here.)
No early final exams, according to University rules.
If you have another final exam scheduled at the same time
as our final exam,
you need to give me the details
(course number and instructor)
by Monday of the 7th week of classes.
Similar comments on the content apply as for the midterm.
The final exam from the last time I taught the course is
here
(pdf);
LaTeX.
Its solutions are
here
(pdf);
LaTeX.
I keep originals of the final exam, but will give out on request
a scan or copy of your exam.
Learning Disabilities
Students with documented learning disabilities who wish to
use the
Accessible Education Center
to
take
tests under specifically arranged conditions
should let me know as soon as possible,
certainly by Wednesday of the third week of classes.
Such students must also
be sure to meet the Accessible Education Center's
separate deadlines for requests;
these are likely to be a week or more in advance of the exam date
(much more for final exams).
I can't do anything to help a student who misses its deadline.
(I have tried in the past.)
Grading Complaints
Complaints about the grading of any exam must be submitted in
writing by the beginning of the first class period after the class in
which that exam is returned.
Extra Credit
Extra credit will be awarded to the first person to detect any
error or misprint in the current edition of the book,
in homework solutions, in exams and exam solutions,
or on the course website.
More extra credit will be given for catching mathematical errors than
misspelled words or wrong dates.
To get the extra credit, you must tell me what the correct version
should be.
There are likely to be a number of mistakes to catch.
When going through materials from the last time I taught
this course,
I found a number of misprints,
including a few egregious ones,
which were not caught despite a similar offer for extra credit.
(A particularly bad one in a homework solution
for the fall quarter was
"not compact" for "compact".)
The corresponding part of the
pdf
version of the syllabus
lists a few which have already been identified,
so are not eligible for extra credit.
Learning Outcomes
The course deals with:

Differentiability of functions, differentiable functions,
and their derivatives,
for functions with domain a suitable subset of R
and codomain R^n or C^n.

Riemann and RiemannStieltjes integrability of functions,
integrable functions,
and their integrals,
for functions with domain an interval in R
and codomain R^n or C^n.

Pointwise and uniform convergence of sequences and series of functions
with domain a metric space
and codomain R^n and C^n.
Continuity, integrability,
and differentiability of the limit function
under suitable hypotheses on the type of convergence,
domain of the function,
and the functions in the sequence or series.

Special properties of power series.

Equicontinuity and the ArzelaAscoli Theorem.

The StoneWeierstrass Theorem for compact metric spaces.

Rigorous construction of the standard functions found in
elementary calculus courses,
together with rigorous proofs of their properties
(continuity, calculation of derivatives,
functional equations, etc.).

Some basic properties of Fourier series.
The successful student will be able to
precisely state the definitions associated with these topics,
and will be able to rigorously prove standard facts.
The successful student will also be able to
rigorously prove or disprove (as appropriate)
statements about these topics which have not been encountered before,
and which are at a level of difficulty
appropriate to a midlevel introductory analysis course.
The successful student will be able to write these proofs
so that they are correct, complete, clear, readable,
and in logical order.
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Academic Conduct
The code of student conduct and
community standards is
here.
In this course, it is
appropriate to help each other on homework as long as the work you are
submitting is your own and you understand it,
and
you give the names of any people you cooperated with.
It is not appropriate
to help each other while taking exams,
to look at other students' exams while taking exams,
or to
bring any unauthorized material to exams.
The only authorized material is pens and pencils.
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Schedule
We will cover most of Chapters 58 of the book in the winter quarter,
possibly getting into Chapter 9.
The part of Chapter 5 already done in the previous course
(going through L'Hospital's Rule) won't be repeated.
We will follow the book fairly closely.
The book usually used in the preceding course is
S. Abbott, Understanding Analysis.
Much of the material of this quarter is also there,
but often restricted to special cases.
As a good example from last quarter,
continuity is mostly only treated there in the context of
functions from subsets of the reals to the reals,
and we will consider continuity of functions
between metric spaces.
and we considered continuity of functions
between metric spaces.
This quarter,
we will often take the codomain for differentiable
and integrable functions to be R^n
or C^n,
and we will consider the RiemannStieltjes integral
instead of just the Riemann integral.
The schedule gives approximate times
that topics will be discussed.
It is necessary to read the book; not everything will be done
in class.
Homework assignments are subject to change.
Week 1 (711 Jan.)
Finish Chapter 5 (differentiation).
Start Chapter 6 (RiemannStieltjes integration).
Homework (due Monday 14 Jan.):
Chapter 5: 1, 2, 3, 4, 5, 9, 11, 13.
Point values: 10 points per problem, except 4 points per part
in Problem 13
and 12 points for Problem 2.
Correction in Problem 13: it should be  x ^a in the definition
of the function f.
(See the
pdf
version of the syllabus.)
Week 2 (1418 Jan.)
Continue Chapter 6 (RiemannStieltjes integration).
Homework (due Wednesday 23 Jan.):
Chapter 5: 12, 22ac, 26; Chapter 6: 2, 4; Problem A.
Point values: 10 points for each problem part.
(Chapter 5 Problem 22ac and Problem A have 3 parts each.)
Problem A:
see the
pdf
version of the syllabus.
Week 3 (2124 Jan.)
Continue Chapter 6 (RiemannStieltjes integration).
(Short week, due to holiday Monday.)
Homework (due Monday 28 Jan.):
Chapter 6: 7, 8, 10, 11.
In Problem 10d,
do only the improper integrals from Problem 7.
Point values: 10 points for each problem part.
(Problem 7 has 2 parts and Problem 10 has 4 parts.
Total 80 points instead of 100 points.)
Week 4 (28 Jan.1 Feb.)
Finish Chapter 6 (RiemannStieltjes integration);
start Chapter 7 (sequences and series of functions).
Homework (due Monday 4 Feb.): Chapter 6: 15;
Chapter 7: 1, 2, 3, 4.
The notation in Problem 15 is not very good;
see the
pdf
version of the syllabus.
Point values: 20 points per problem.
Week 5 (48 Feb.)
Continue Chapter 7 (sequences and series of functions).
Homework (due Friday 8 Feb.; short assignment,
because of the midterm on Monday):
Chapter 7: 5, 6, 7.
Point values: 13 points each for Problems 5 and 6;
14 points for Problems 7.
(Total 40 points instead of 100 points.)
Review session for the midterm:
Friday 6 February 57 pm, 205 Deady.
Week 6 (1115 Feb.)
Midterm Monday 11 Feb.,
68 pm, 205 Deady.
Finish Chapter 7 (sequences and series of functions).
Homework (due Monday 18 Feb.):
Chapter 7: 9, 16, 19, 20, 21.
The notation in Problem 20 is not very good;
see the
pdf
version of the syllabus.
Point values: 20 points per problem.
Week 7 (1822 Feb.)
Start Chapter 8 (standard elementary functions; Fourier series).
Homework (due Monday 25 Feb.):
Chapter 7: 22, 24;
Chapter 8: 2, 3, 4, 5.
(In Chapter 7 Problem 22,
you may use Problems 6.11 and 6.12.)
Point values: 5 points per part in Problems 4 and 5;
otherwise, 10 points for each of the two other
problems in Chapter 8,
and 20 points for each of the problems in Chapter 7.
Week 8 (25 Feb.1 March)
Continue Chapter 8 (standard elementary functions; Fourier series).
Homework (due Monday 4 March):
Chapter 8: 6, 7, 8, 9, 10, 11.
(There is a hint for Problem 10, but it is on the page
after the statement of the problem.)
Point values: 10 points for each problem part,
except 15 points each for Problems 8 and 11
and 20 points for Problem 10.
Week 9 (48 March)
Continue Chapter 8 (standard elementary functions; Fourier series).
Homework (due Monday 11 March):
Chapter 5: 27; Chapter 8: 20, 23, 24, 25.
(Also look at Problems 22 and 26.)
Point values: 20 points per problem.
Week 10 (1115 March)
Finish Chapter 8 (standard elementary functions; Fourier series).
Homework (due Friday 15 March):
Chapter 8: 12, 13, 14.
Point values: 10 points for each problem part.
(Total 70 points instead of 100 points.)
Sa 16 March, 2:004:00 pm: Review session for the final exam.
Room to be announced.
Final exams week (1822 March)
M 18 March, 10:15 am12:15 pm: final exam,
in regular classroom.
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Publicly Available Documents Associated with this Course
Here is a list of publicly available documents
associated with this course.
The material is arranged in approximate chronological order: most
recent items at the bottom.
Links to written homework solutions,
exams, and exam solutions will not work until after
the corresponding written homework has been turned in
or the corresponding exam has been given, and
the links to sample exams and their solutions will not work until
these items have been prepared.
Most files will be posted in pdf and AMSLaTeX versions.
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Important dates, according to the
Academic
Calendar at the registrar's office
(not guaranteed!)

Su 6 January:
Last day to process a complete drop (100% refund, no W recorded).

Sa 12 January:
Last day to drop this course (100% refund, no W recorded;
after this date, W's are recorded).

Sa 12 January:
Last day to process a complete drop (90% refund, no W recorded).

Su 13 January:
Last day to add this course.

Su 13 January:
Last day to withdraw from this course (100% refund, W recorded).

W 16 January:
Last day to change to or from audit.

Su 20 January:
Last day to withdraw from this course (75% refund, W recorded).

M 21 January:
No classes.

Su 27 January:
Last day to withdraw from this course (50% refund, W recorded).

Su 3 February:
Last day to withdraw from this course (25% refund, W recorded).

Su 24 February:
Last day to withdraw from this course (0% refund, W recorded).

Su 24 February:
Last day to change grading option or variable credits for this course.
(Tuition penalties apply when reducing credits.)
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This page maintained by
N. Christopher Phillips,
email.
Please email plain text
(7 bit ASCII)
only
(no web page coded files, Microsoft Word documents, binary
characters, etc.; see above for more).
Last significant change: 7 January 2019.