MAT337

Updates:

April 27: At a student comment, I corrected a typo and clarified the solution to
Problem 7.3 F, part a), in Assignment 5.

April 27: At a student's request, I added the solution to Problem 10.10 D in Assignment 8.

April 26: A few typos were corrected, and further clarifications were added in the
solution to Problem 4) in Assignment 4). This was prompted by a student's question,
as is the following correction: in class I wrote a small comparsion table saying that
Q is both "nowhere dense, and is dense" in R, and so is the space of all polynomials in
C([a,b]). Of course this should be replaced by "has empty interior, and is dense"
(for both examples). The "empty interior" claim follows from Baire's theorem, as these
sets are countable unions of nowhere dense sets. For the polynomials - the subspaces
of polynomials of degree smaller or equal to n are the nowhere dense sets.

April 26: I will be in the office tomorrow (Wednesday) at 2:30 for an office hour.

April 25: Fourth note of the day: I will try to be in the office tomorrow (Tuesday) at 12:30 for an office hour.

April 25: Third note of the day: I will be in today (Monday) at 4:30 for an office hour.

April 25: Second note of the day:
The final exam structure: as indicated in class, the exam will have 7 questions
and will cover the material cumulatively, including material from both before and after
the term tests. As usual, you should not memorize proofs of long theorems, but
you do have to know the definitions and results covered in the course. Some parts of
questions may ask to quote these. Most of the questions will have the form of problem
solving, and some of them will have guidance in the form of hints. You should therefore
try to go over as many problems as possible, both from the assignment and from
other places in the book and elsewhere. Make sure you have covered problems from
each of the main topics, as outlined from the course notes. In the book they are given
as follows:
Metric spaces - basic properties, compactness, Cauchy sequences and completeness (Ch. 9,4);
Normed vector spaces - Banach spaces, spaces of functions and their subspaces
(applications of metric space theory to this case), (Ch. 7);
Inner product spaces - Hilbert spaces, orthogonality, orthonormal sets and bases (Ch. 7);
Continuity - extreme values of continuous functions, uniform continuity (Ch. 5);
Limits in function spaces - uniform convergence and associated topics (Ch. 8);
Approximations results - culminating in Stone-Weierstrass (Ch. 10, Sections 1,2,10);
Baire Category - (Ch. 9, Section 9.3);
The Contraction principle (Section 11.1).
Of course this list cannot replace the detailed ones given before (see below), but it
does provide a bird's-eye view of the material.

April 25: As I described in class, the material for the final exam is everything
taught in the course except the last two hours, i.e measure theory and connectedness
are not included.
To spell this out positively, see the previous notes (Feb. 2 and March 14) for
the early material. The part of the exam material that appeared after the term tests is:
limits in function spaces (convergence - pointwise, uniform and in other norms;
completeness of C([a,b]) and of more general spaces of continuous functions, uniform
convergence and intgeration/differentiation, differentiation under the integral sign,
series - uniformally convergent, Cauchy uniformally convergent, the M-test),
approximation by polynomials (Taylor, Weierstrass and Stone-Weierstrass Theorems),
Baire Category Theorem (and related concepts: nowhere dense sets, sets of 1st category,
different formulations of the theorem, closed set version of the Cantor Intersection
Theorem), the Banach contraction Principle.

April 19: Solutions to Assignment 8 are now available.

April 18: Solutions to Assignment 7 are now available. I will be holding an office
hour on Wednesday, April 20, at 12-1 p.m. At that time, you should be
able to pick up Assignment 8 (I hope they will be ready by then). After that
time the assignments will be available in an envelope near my office.

April 9: Assignment 8 has been extended to Tuesday, April 12 at 12 p.m. It can
be handed in at my office (SS4091) if I am there, or at the Math office on the
4th floor of SS, in an envelope addressed to me. Remarks on some of these problems
can be found here.

April 1: Midterm 2 average was a little over 31/60. Its solutions are posted below
in five .jpg files.

March 31: Assignment 8 has been posted, due April 8.

March 29: Assignment 7 has been postponed, and is now due on Friday, April 1.
Assignment 8 will be due on April 8.

March 18: Solutions to Assignment 6 are now posted.

March 16: Solutions to Assignment 5, as well as Assignment 7 itself, are now posted.
Assignment 7 is due on March 30 in class.

March 14: Material for term test II: Cauchy sequences and completeness, total
boundedness, normed vector spaces (basic properties and examples (L^p, l^p,
the infinity norms, etc.)), Banach spaces, continuity of functions on metric spaces,
Weirstrass and extreme value theorems, continuity of the norm function, linear
operators between normed vector spaces (especially bounded ones, with the norm
on those), inner product spaces (basic properties, Cauchy-Schwarz inequality, Hilbert
spaces, continuity of the inner product, the parallelogram law), orthogonality
(definition, the orthogonal complement, orthogonal set, orthonormal bases, the
Gram-Schmidt process, Fourier coefficients), orthogonal projections in Hilbert
spaces (for closed convex sets, the main orthogonal projection theorem, consequences
for projections on finite dimensional subspaces, Bessel's inequality, Parseval
identity for separable Hilbert spaces), l^2 as a Hilbert space, finite dimensional
normed vector spaces (topology, completeness, existence of projections onto a finite
dimensional subspace), uniform continuity.

March 14: A link to scan of Term-Test 1's solution appears below (in 3 parts).

March 13: Problem 7.5G is more involved than others. Here are a few points and hints
about it.

March 13: Assignment 6 is postponed to Wednesday, March 16. However, Assignment 7
will be posted on Monday, March 14 and will be due on Wednesday, March 30, a week
after the term test. There will also be an Assignment 8 which will be due on Wednesday,
April 6. Later today I will post some hints on one of the problems in Assignment 6.

March 11: Solutions to problem set 4 are now posted.

March 11: In Assignment 6, Problem 7.5G, the definition of M + N is
{v+w | v is in M and w is in N}, for any two sets M,N in a vector space.
This should be familiar from Linear Algebra. In Problem 7.6F, there are
some typos: V is just R^n (with the infinity norm), and the unit ball referred
to is the closed one (all x with R^n whose infinity norm is less than or equal to 1).
Thanks to the student who pointed this out.

March 7: Assignment 6 is now posted.

Feb. 23: Assignment 5 is now posted.

Feb. 22: Assignment 4's deadline has been extended to Friday, February 25 in class.
Clarifications on the assignment: in problem 4, the phrase in bracket is misleading -
equivalence is meant in the sense of part of the conclusion of problem 9.1Gb -
having the same convergent subsquences. Also, in 9.1Mb), add "decreasing" after
"every". Both these are now corrected in the assignment. Thanks to the students
who pointed this out.

Feb. 10: More detailed versions of the solutions to assignments 2,3 are now posted.
Note that at 1)J)c) of assignment 2 there are a few places with a sign missing in the
power: e.g. 2^(-[n/2]) --> 0.
With the new solutions to the assignments posted, good luck on the test tomorrow.
Here are a few more points about it: it will take place during class time and in
class (SS1070). Come early and bring your student id! The test contains 5 problems.
In one of the problems you are asked to do just one of two parts. In the material list
I posted (Feb. 2), you can put less emphasis in studying the first and last topic - and
in general, not every topic is covered in such a short test.

Feb. 7: Solutions to Assignments 2 and 3 are now posted.

Feb. 6: Assignment 4 is now posted, due on Wednesday, February 23, after the break.

Feb. 2: After a unanimous vote in class, term test I has been scheduled to
Friday, February 11, 2005 (note the change!)
The material for the test:
Cardinality, definition of a metric space and examples, open balls, open and closed
sets, the Hausdorff property, sequences, limits and limit points, closure of a set
and its characterizations, compactness, bounded sets, the Bolzano-Weierstrass
Theorem and the Heine-Borel theorem, the finite intersection property, sequential
compactness and the Borel-Lebesgue theorem, the interior and boundary of a set,
relative metric spaces, nowhere dense sets and the Cantor set, basic properties of
Cauchy sequences.
Completeness is NOT a topic in the test.
Corresponding book sections: 2.8, Chapter 4 except the last part of 4.2, Sections 9.1
and 9.2 except their very end, and just the beginning of definition 9.3.1.
Work on solving problems, know the definitions, and principles. There is no need to
memorize long and complicated proofs of the main theorems.

Jan. 28: Issues related to Assignment 3: In Section 9.2, It appears that the
symbol U^c in the hint of problem E is the complement of U in R^n, even though the
book often uses another convention.
In problem 3 there are some misleading issues: the finite intersection property can be
defined for any collection of sets. However, in class we have defined and used it only
for closed sets. This caused me to initially neglect writing the word "closed" in Problem 3.
That, in turn led to another confusion: are the sets C_i's, which are in K,
closed in X, or closed in K with its relative metric. If the former - then my hint
is not necessary, since the proof in class works with only minimal changes. My hint
therefore, is meant for the latter case, but any interpretation you give will be
honored by the grader, but be explicit about which one you use. I hope this settles
the matter.
Terms, such as "inf" and "sup" (see Section 4.4 problem I, for example) should be
known from previous courses, but perhaps with a different name: "inf" is the greatest
lower bound and "sup" is the least upper bound, defined for sets of real numbers.
Solutions: an updated version for solutions to assignment 1 is now posted. It
has a few typos in problem 3 (missing curly brackets and the empty set in the definition
of a finite set) which are inconsequential.
There may be further updates later in the weekend.

Jan. 27: The teaching assistant has made available some solutions to Assignment 1. They
are linked in the assignments web-page. While these are printed, due to lack of time,
later solutions may only be given in hand-written form.

Jan. 24: Assignment 3 is available below, due Jan. 31.

Jan. 13: Handing in Assignment 1 has been postponed to Monday, Jan. 17 at 12p.m.
(in the beginning of class). Also, another hint for problem 4.: combine the given
hint with a proof analogous to the way it was shown in class that |[0,1)|=|(0,1)|.

Jan. 12: A large number of copies of the course textbook is finally available at
the university bookstore.

Jan. 11: Another student pointed similarly a typo in problem 5., where an extra
square appears in the Pythagorean identity. This, along with a wrongly placed
colon in that problem are now corrected.

Jan. 10: As pointed out by a student, Problem 3. in Assignment 1 preserves an
omission in the book's Problem K of Section 2.8, as one wants the set A to be
countable and also infinite (so, not finite). This has now been corrected in the
problem set.

Jan. 7: Assignment 1 is finally available on the link below, and due Jan. 14.

Links to the course outline, assignments and solutions and to remarks on the
book versus the lectures. Assignments will be updated weekly.