18798-MATH 510 001 Intro To Analysis.

Class time  10:00 am-10:50 pm. Location MITCH-202

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Instructor: Dimiter Vassilev     Office : SMLC 326  Email: vassilev@unm.edu  Phone Number: 505 277 2136

 

Office Hours: MWF 1:30pm-2:30pm

 

Final Exam: Friday, December 16, 7:30am-9:30am.

Students having conflicts with this examination schedule must notify the appropriate instructor before Friday, November 14, 2016. Any student having more than three examinations scheduled in any one day may notify the instructor of the last examination listed. If notified before November 14, 2016, the instructor shall make arrangements to give a special examination. Conflicts arising as a result of scheduling out of normal hours-pattern or day sequences must be resolved by the instructor of the off-pattern courses.        

 

 

Texts: class notes based mainly on the following optional texts

 

Please note the following guidelines for the course:

 

Course content: This is a first graduate course on Analysis. It is also the course that prepares graduate students for the Real Analysis Qualifying. Students should have at this point enough computational background (calculus of one variable and multivariable calculus -at least 2 and 3 variables), and I expect most of you to have been exposed at least once to rigorous epsilon-delta proofs, if not you might consider taking Math 401/501 instead. I expect familiarity with the real and complex numbers.


Description: Real number fields, sets and mappings. Basic point set topology, sequences, series, convergence issues. Continuous functions, differentiation, Riemann integral. General topology and applications: Weierstrass and Stone-Weierstrass approximation theorems, elements of Fourier Analysis (time permitting).

 

Grades: The final grade will be determined by homework (25%), two midterms( 50%) and a final exam (25%). All grades will be posted on UNMLearn.

 

Homework  There will be one weekly homework. You can work together on the homework, but you do need to write up your own solutions in your own words. To help the grader, please write your solutions up neatly and clearly (no points for work that the reader cannot follow- this is true also for exams), and staple the sheets.   The two lowest homework grades will be dropped. Please no late homework!  Problems from past real Analysis Qualifying exams will be weaved into the homework, hopefully by the end of the course you will have built a folder with solutions to most of those problems for future reference.

 

 

 

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; BVloc(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 Φp(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 - Ck, Ck,α, Rp - 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