5/6/05: Discussion of the sample final. [Posted on the class web page]
5/6/05: Discussion of the sample final. [Posted on the class web page]
5/4/05: Convergence in the mean: proofs of the Riesz-Fischer theorem
and Parseval's Identity. The existence and uniqueness of an
L^2-periodic function with given Fourier coefficients. [Class notes;
S: Sections 12.2.3, 14.4.3, and 12.2.2]
5/2/05: Another Weierstrass approximation theorem. Convergence in the
mean: the Riesz-Fischer theorem, Parseval's Identity. (No proofs yet.)
[Class notes; S: Sections 12.2.2 and 14.4.3]
4/29/05: Fejer's theorem. [Class notes; S: Section 12.2.2]
4/27/05: Relation between L^2- and L^1-periodic functions. Bessel's
inequality. Dirichlet's theorem. [Class notes; S: Sections 12.2.1 and
12.2.3]
4/25/05: Fourier expansions and coefficients. The Riemann-Lebesgue
lemma. [Class notes; S: p. 522, Section 12.2.3]
4/22/05: Fourier series: introduction. [Class notes; S: pp. 531-532]
4/20/05: f = 0 implies \int f = 0 (proof). The spaces L^1 and
L^2. Correction: As the homework indicates, L^1 intersected
with bounded functions is a subspace of L^2. I forgot the boundedness
condition when I talked about that in class. [Class notes; S: Sections
14.4.1 and 14.4.2]
4/18/05: Convergence theorems: monotone, Fatou. "Almost everywhere."
[Class notes; S: Sections 14.3.2 and 14.3.4]
4/15/05: The dominated convergence theorem. [Class notes; S: Section
14.3.3]
4/13/05: More on integrability. Convergence theorems: a monotone
convergence lemma. Read the wordings of the convergence theorems on
your own, skip the proofs for the time being. Note that we define
Lebesgue integral as the supremum of the integrals of simple
functions, rather than using Lebesgue approximate sums, which are
concrete examples of approximations by simple functions. Read also the
notion of "almost everywhere" and the wording of the theorem about the
integrals in that section. [Class notes; S: Sections 14.3.2, 14.3.3,
and 14.3.4]
4/11/05: Simple functions. Lebesgue integration: simple and measurable
functions. [Class notes; S: Sections 14.3.1 and 14.3.3 (before Theorem
14.3.5)]
4/08/05: The second midterm test.
4/06/05: Review before the test.
4/04/05: Properties of measurable functions. [Class notes; S: Section
14.3.1]
4/01/05: A nonmeasurable set. Measurable functions. [Class notes; S:
Section 14.3.1]
3/30/05: Fundamental properties of measurable sets. Lebesgue
measure. [Class notes; S: Section 14.1.4]
3/28/05: Properties of outer measure. Measurable sets: first
properties. [Class notes; S: Sections 14.1.4 and 14.1.5]
3/25/05: HW discussion. Outer measure. [Class notes; S: Sections
13.1.3, 14.1.2, and 14.1.5; Rudin: pp. 302-304]
3/23/05: An example on extrema with constraints. Lebesgue measure:
motivation. The intuitive notion of volume. The Banach-Tarski
paradox. [S: Section 14.1.1]
3/21/05: Extrema with constraints. Lagrange multipliers. [S: Section
13.3.1]
3/14-20/05: The Spring Break.
3/11/05: HW discussion. The Implicit and Inverse Function
Theorems. [S: Sections 11.1.6 and 13.1.1]
3/9/05: The proof of the existence and uniqueness theorem. Linear
differential equations and systems. Read on Picard's iterations for
higher-order DEs on your own. [S: Sections 11.1.2, 11.1.3, and 11.1.5]
3/7/05: ODEs: Picard's iterations, existence and uniqueness. [S:
Section 11.1.2]
3/4/05: Your graded exams and solutions handed out and discussed. HW
collected and discussed. Ordinary differential equations (ODEs):
introduction. [S: Sections 10.2.4 and 11.1.1]
3/2/05: Commutation of partial derivatives (brushed up). Taylor's
formula (proof). Extrema (left for your own study). [S: Sections
10.2.3 and 10.2.2]
2/28/05: Commutation of partial derivatives. Taylor's formula (statement). [S:
Sections 10.2.1 and 10.2.3]
2/25/05: Midterm exam. [S: see coverage under News and Announcements]
2/23/05: Partials and the differential. Directional derivatives. The
chain rule. [S: Sections 10.1.2 and 10.1.3]
2/21/05: Differentiating integrals, i.e., integrals depending on
parameters. The differential. [S: Sections 10.1.4 and 10.1.1]
2/18/05: HW discussion. Partial derivatives. [S: Section 10.1.2]
2/16/05: Connectedness. The Banach Fixed Point theorem (i.e., the
Contractive Mapping Principle). [S: Sections 9.3.3 and 9.3.4]
2/14/05: Continuity and compactness. [S: Section 9.3.2]
2/11/05: HW discussion. Continuous functions: definitions and general
properties. [S: Section 9.3.1]
2/9/05: Closed sets (finished): complements, unions, and
intersections. Compactness and completeness. [S: Sections 9.2.2,
9.2.3, and 9.2.4]
2/7/05: Open sets (finished): int A is always open and the largest
open set contained in A; intersections and unions of open sets. Closed
sets (begun): limits in metric spaces, limit points, and closed
sets. [S: Sections 9.2.1 and 9.2.2]
2/4/05: HW discussion. Open sets. [S: Section 9.2.1]
2/2/05: Metrics, norms, and inner products. [S: Sections 9.1.1, 9.1.2,
9.1.3]
1/31/05: The exponential function and the logarithm. [S: Section
8.1.1]
1/28/05: Homework discussion.
1/26/05: Equicontinuity and the Ascoli-Arzela Theorem. [S: Sections
7.6.1 and 7.6.2]
1/24/05: The Weierstrass Approximation Theorem (proof). [S: Section
7.5.3]
1/21/05: Convolutions (continued), approximate identity. [S: Section
7.5.2]
1/19/05: Lagrange interpolation, the Weierstrass Approximation Theorem
(formulation), and convolutions. [S: Sections 7.5.1, 7.5.2 (through
p. 298), and 7.5.3 (omit the proof so far)]
12/15/04: Discussion of the sample final exam: Problems 2, 4. I am
posting solutions to the other ones.
12/13/04: Discussion of the Fall 2000 final exam: Problems 1,
3. Problem 2 on the copy handed out in class uses something that will
be covered next term only and is therefore irrelevant. I have changed
it to a relevant one on the posted version.
12/10/04: Discussion of the homework: Problems 1 and 6 on p. 294.
12/08/04: The uniqueness of the coefficients of a convergent power
series (the formula through the higher derivatives of the resulting
function). Analytic functions, analytic continuation. [S: Sections
7.4.1 (p. 281) and 7.4.2]
12/06/04: Power series, the radius R of convergence, the uniform
convergence on closed subintervals of (-R,R). Differentiability (study
on your own). [S: Section 7.4.1]
12/03/04: Limits of function sequences and integration and
differentiation. [S: Section 7.3.2]
12/01/04: Discussion of the homework.
11/29/04: Uniform convergence. The continuity, integrability, and
differentiability of the limit. [S: Sections 7.3.1 and 7.3.2 (skipping
the proofs in 7.3.2 for the time being)]
11/24/04: Series. The divergence of the harmonic series. The Cauchy
criterion for series. Absolute convergence. Rearrangements and
conditional convergence. [S: Sections 7.2.1 from p. 254 and 7.2.2]
11/22/04: Discussion of the homework problem (#1 on p. 235) on
improper integrals: correction of what we did last time. Read the
first 3 1/2 pages of the section on convergence of series on your own,
as most of it is usually done pretty much the same way in calculus,
and think about questions to ask me on Wednesday about that
material. [S: Section 7.2.1 through p. 253]
11/19/04: Discussion of the homework, problems on improper
integrals. Read the section on complex numbers and complex-valued
functions on your own, as most of it must be just a review for
you. [S: Section 7.1]
11/17/04: Discussion of the exam. Improper integrals. [S: Section 6.3]
11/15/04: The Riemann integral: proving the oscillation
test. Elementary properties of the integral. [S: Sections 6.2.1 and
6.2.2]
11/12/04: The Riemann integral. [S: Section 6.2.1]
11/10/04: Midterm exam II.
11/08/04: If the function is continuous and bijective on (a,b), then
it is monotone. The integrability of a continuous function on a closed
bounded interval. A sample problem on (continuous) differentiability
x^3 sin (1/x). [S: Sections 6.1.1 and 6.1.2]
11/05/04: Problem 23 on p. 194. The integral of a continuous
function. The Cauchy and the upper and lower Riemann sums. The
fundamental theorem of calculus. [S: Sections 6.1.1 and 6.1.2]
11/03/04: Higher derivatives and Taylor's Theorem. Using the second
derivative to study concavity and extrema of functions. The second
difference (left for your own reading and enjoyment). [S: Sections
5.4.1 and 5.4.2]
11/01/04: The inverse function theorem. [S: Section 5.3.3]
10/29/04: Discussion of Problem 8 on p. 165. Useful tool, lemma
providing a test for differentiability: f(x) - f(x_0) = phi(x) (x-x_0)
with phi continuous at x_0. Using this lemma to prove the Chain
Rule. Using the Chain Rule to prove the Quotient Rule. [S: Section
5.3.2 and Rudin: Theorem 5.5, which is proven practically the same way
we did it]
10/27/04: Global properties of the derivative: positive derivative and
increasing functions. The Intermediate Value Theorem. The calculus of
derivatives: the sum, difference, product, and quotient rules. [S:
Sections 5.2.2, 5.2.3, and 5.3.1]
10/25/04: Properties of the derivative: extrema and the Mean Value
Theorem. [S: Sections 5.2.1 and 5.2.2]
10/22/04: Derivative. The "oh"s. [S: Section 5.1]
10/20/04: Solving problems on continuous functions. [S: Section 4.2]
10/18/04: Properties of continuous functions, before monotone
functions. [S: Sections 4.2.1 and 4.2.2]
10/15/04: Functions, continuity, limits. [S: Section 4.1]
10/13/04: Discussion of the first midterm. Proof of Theorem 3.3.2
(Heine-Borel's Lemma) [S: Section 3.3, study the nested compact sets
on your own]
10/11/04: Compact Sets. [S: Section 3.3]
10/08/04: Midterm Test 1.
10/06/04: Discussion of Problems 14 and 5 from pp. 98-99. [Strichartz:
Section 3.2.3.]
10/04/04: More about closed sets. The interior, the closure, and the
boundary. [Strichartz: Section 3.2.2.]
10/01/04: Open and closed sets. [Strichartz: Sections 3.2.1 and
3.2.2.]
09/29/04: Limit points of a sequence. Limsup and liminf. [Strichartz:
Section 3.1.2.]
09/27/04: Limits, suprema, and infima. Every bounded above set has a
supremum. Every increasing sequence bounded above has a
limit. [Strichartz: Section 3.1.1.]
09/24/04: Other ways to construct the system of real numbers. Infinite
decimal fractions. [Strichartz: Section 2.4.1.]
09/22/04: Square roots. [Strichartz: Section 2.3.2.]
09/20/04: Order. The axioms of an ordered field. Limits and
completeness. [Strichartz: Sections 2.2.3 and 2.3.1.]
09/17/04: Arithmetic with the reals: why the product is
well-defined. The axioms of a field. [Strichartz: Sections 2.2.1 and
2.2.2.]
09/15/04: Discussion of homework. The rationals as part of our real
number system. Arithmetic with the reals. [Strichartz: Section 2.2.1.]
09/13/04: The system of real numbers: expected properties (axioms) and
different way of constructing it. The definition of a real number as
an equivalence class of Cauchy sequences. Digression about equivalence
relations. [Strichartz: Section 2.1.]
09/10/04: More on countable and uncountable
sets. Cardinality. Cantor's diagonal argument. The uncountability of
the set of real numbers. [Strichartz: Section 1.2.2. Also read the
section on rational numbers.]
09/08/04: What is Analysis and what is this class about? Injective,
surjective, and bijective functions. Countable and uncountable
sets. [Strichartz: pp. xiii-xv (Preface) and Section 1.2.1]