Class Day

Topic

Homework due Wednesday the following week

1  M  Aug 22 

 The real numbers as cuts

HW1

2  W Aug 24 

 sup and inf; Archimedean property; denseness of the rational numbers; limit points; open and closed sets; sequences

3   F Aug 26

limsup and liminf. The Bolzano-Weierstrass' thrm. Cauchy sequences, completeness of R.

4   M Aug 29 

 Limits of functions, continuous functions- basic properties.

HW 2

5   W Aug 31 

Tests for absolute convergence of series. The number e.

6   F  Sep 2 

Cauchy test for convergence. Nested intervals. Continuous functions - IVP, min/max on a closed bounded interval

Sep 5 Labour Day

7   W Sep 7 

 Limit of a function using sequences; continuity of the inverse function

HW 3

8   F Sep 9 

 Uniform convergence; uniform convergence of continuous  or bounded functions; Weierstrass' M-test for series;

See also Classes #41- #43.

9   M  Sep 12 

Leibnitz' test for convergence of series of numbers and uniform convergence of series of functions

HW 4

10  W Sep 14 

 (Sequential) compactness; Heine-Borel' theorem; product of series one of which is absolutely convergent

11  F Sep 16 

 Exponential and logarithmic functions; Merten's theorem

12  M Sep 19 

 Abel's theorem; applications

HW 5

13  W Sep 21 

 Summability - Cessaro (arithmetic mean) & Abel (geometric mean); see also Class #40.

Tauberian theorems. The binomial formula

14  F Sep 23 

The derivative: Rolle's theorem and the MVT, Darboux's theorem and discontinuities of the derivative;

uniform convergence and the derivative - derivative of the exponential function, derivative of uniformly

convergent series of function

15  M Sep 26 

Proof of the interchange of uniform limit and differentiation;  Cauchy's MVT; l'Hopital's theorem.

HW 6

16  W Sep 28 

Derivative of inverse and composition; Taylor's formula; analytic functions; the Weierstrass approximation theorem-

see also Class #45.

17  F Sep 30 

Proof of Weierstrass' theorem; uniform continuity; equicontinuous families and point-wise bounded sets of functions

18  M Oct 3 

Ascoli's theorem; see also Class #45.

HW 7

19  W Oct 5

Poincare recurrence theorem; "irrational" rotations on the circle; Nα+Z is dense in R

20  F Oct 7 

 Exam 1 (Exam 1 solutions)

21  M Oct 10 

Continue  "irrational" rotations on the circle and  Nα+Z is dense in R (Dirichlet's theorem)

22  W Oct 12 

Finish the proof of Dirichlet's theorem; Example of a continuous nowhere differentiable function

               Oct 14 FALL BREAK

23  M Oct 17 

The Riemann-Stieltjes integral - F. Riesz' theorem. Functions of bounded variations-basic properties and Jordan's theorem

HW 8

24  W Oct 19 

 Proof of Jordan's decomposition; continuity properties of the variation; positive and negative variation

25  F Oct 21 

 Relation between the positive/negative variation and the variation; BV_loc(R); BV(R);

continuous function which is not in BV([a,b])

26  M Oct 24 

 The Riemann-Stieltjes integral. Basic properties (linearity); reduction to Riemann integral; examples

HW 9

27  W Oct 26 

 Additivity of the RS integral w.r.t. the domain (sufficient condition); Existence of the RS integral w.r.t. a BV function for

continuous functions.

28  F Oct 28 

 Sufficient condition for non-integrabilty; other definitions of Riemann-Stieltjes integrals;

Integration by parts; The R-S integral as a continuous linear functional f-->∫ fdg

29  M Oct 31 

 Proof of the F. Riesz' theorem

HW 10

30  W Nov 2 

 Limits of the R-S integral  ∫ fdg w.r.t. g. Helly's theorems. The fundamental theorem of calculus for R-S integral.

31  F Nov 4 

 MVT for the Riemann-Stieltjes  integral; change of variables; Helly's selection theorem (compare with Ascoli's theorem);

other definitions of the R-S integral

32  M Nov 7 

 Recovering the function from its derivative

HW 11

33  W Nov 9 

 The Riemann-Stieltjes integral for bounded function w.r.t. an increasing function using Darboux-Riemann sums;

The Riemann integral - properties.

34  F Nov 11 

The Lebesgue theorem - sufficient and necessary condition for Riemann integrability.

35  M Nov 14 

 Arzela's (bounded) dominated convergence theorem. Improper integrals - comparison tests, Dirichlet's test.

HW 12

36  W Nov 16 

 Dominated convergence theorem for improper integrals. Convex functions - geometric properties,

continuity and differentiability. Jensen's theorem.

37  F Nov 18 

 Monotonicity of Φ_p(f) . Fourier series - orthonormal systems of functions; Dirichlet's kernel

38  M Nov 21 

 Exam 2 (Exam 2 solutions)

HW 13

39  W Nov 23 

 Dini's convergence theorem. Bessel's inequality; the Riemann-Lebesgue lemma

Nov 25  THANKSGIVING BREAK    

40 M Nov 28 

 Fejer's kernel and convergence; Uniqueness theorem; Weierstrass' theorem. Parseval's theorem. The Poisson kernel.

HW 14

41 W Nov 30 

 Metric spaces - metric balls, open and closed sets, interior, closure, properties

42  F Dec 2 

Examples - C^k, C^k,α, L^p - Minkowski's inequality, equivalence of functions equal a.e..

Limit points - properties, characterization in terms of sequences, perfect sets, the Cantor set

43  M Dec 5

Completeness- examples, normally convergent series, the Contraction Principle, Baire's category theorem

 HW (not collected)

44  W Dec 7

Continuous maps (functions). Separation properties of metric spaces - T₁, Hausdorff (T₂), regular, normal,

separating by a continuous bump function. Compactness in metric spaces, separability, closedness, sequential

compactness (recall Helly's selection & Ascoli' theorems); the Heine-Borel theorem.

45  F Dec 9

Ascoli's theorem; the Stone-Weierstrass theorem

 December 16, 7:30am-9:30am. FINAL EXAM