# MA44000 Real Analysis (Honors)

Purdue University Fall 2011

## Monday, December 12, 2011

### Office hours during the finals week

TTh 12:00-1:00pm, in MATH 610

You may also contact me by email at any time.

## Wednesday, December 7, 2011

### Final Exam

Scheduled Sat, Dec 17, 1:00pm-3:00pm in PHYS 223

Covers all material. We will have review on Wed, Dec 7 and Fri, Dec 9.

New! [Practice Problems] (in addition to Midterm Practice Problems)

Final score

- Your final score is going to be calculated by the following scheme:

Scheme I =(1/3) FE + (7/30) ME1+ (7/30) ME2+(1/5) HW

where FE, MEi, HW are the scores (in %) for Final Exam, Midterm i, Homework

- If you scored above the average on each of the Midterm Exams, that is

if ME1 >= 46 and ME2 >=33

then you will be allowed not to take the final and in that case your score will be calculated by the scheme

Scheme II = (3/8) ME1+(3/8) ME2+(1/4) HW

Note: If you qualify and decide to opt out from taking the final exam, please let me know (by emailing) by Fri, Dec 9.

## Sunday, December 4, 2011

### Homework Assignments

All problems are from [B]
#11 (will not be graded): 35.D(d,e,f), 35.J, 37.B, 37.G, 37.H(c,e,f), 37.J
#10 Due Fri, Dec 2: 31.D, 31.F, 31.K, 34.H, 34.K, 35.I
#9 Due Mon, Nov 14: 29.E, 29.M, 29.S, 30.A, 30.D, 30.F
#8 Due Wed, Nov 2: 25.A(c,d), 25.V, 27.D, 27.H, 28.B, 28.K, 28.O
#7 Due Wed, Oct 26: 23.G, 23.J, 24.C, 24.D, 24.E (a,d,e), 24.N
#6 Due Wed, Oct 19: 18.B, 18.F, 20.G, 20.P, 22.E, 22.F, 22.R
#5 Due Wed, Sep 28: 14.D, 14.O, 15.M, 15.N, 16.A, 16.I, 16.P
#4 Due Wed, Sep 21: 11.A, 11.C, 11.K, 12.A, 12.C, 12.E, 12.F
#3 Due Wed, Sep 14: 9.M, 9.Q(a,b,c,d), 10.C, 10.F, 10.G
#2 Due Wed, Sep 7: 7.J, 8.F, 8.G, 8.M, 8.Q, 9.G, 9.L
#1 Due Wed, Aug 31: 4.H, 5.C, 5.O, 6.D, 6.G, 6.H, 7.G
Note: Problem 5.O has a mistake in it. Correct it and solve it.

## Thursday, December 1, 2011

### Course Log

Covered
- Dec 5: (planned) §37 Power series
- Dec 2: § 35 Ratio test, §37 Series of functions
- Nov 30: §34 Rearrangements of series, §35 Comparison and Limit Comparison tests, Root test
- Nov 18: §31 Uniform Convergence and Integral (completed); Bounded Convergence Theorem, Dominated Convergence Theorem §34 Convergence of Infinite Series, Cauchy criterion, absolute and conditional convergence, nonnegative series.
- Nov 14: §31 Uniform Convergence and Integral
- Nov 11:§30 Differentiation Theorem, Fundamental Theorem of Calculus, Change of Variables
- Nov 7: §30 Riemann Criterion of Integrability, First and Second Mean Value Theorems
- Nov 2: §29 Properties of integral, Integration by parts, Modification of the integral
- Oct 31: §29 Riemann-Sieltjes Integral, Examples.
- Oct 28: §27 Mean Value Theorem, Cauchy Mean Value Theorem; §28 Taylor's Theorem.
- Oct 26: §26 Limit of the function at a point, upper and lower limits; §27Differentiation, Rolle's Theorem.
- Oct 24: §25 Weierstrass approxuamtion theorem (cont.), §26 Limit of the function at a point, upper and lower limits
- Oct 21: §25 Weierstrass approximation theorem (Bernstein's proof)
- Oct 19: §24 Pointwise and Uniform convergence of functions
- Oct 17: §22 Preservation of compactness (cont), §23 Uniform continuity
- Oct 14: §22 Preservation of connectedness, compactness
- Oct 12: §20 Examples, §22 Global Continuity Theorem
- Oct 7: Midterm 1 solutions
- Oct 5: §20 Continuity at a point (topological, metric, and sequential definitions), examples
- Oct 3: Review for Midterm Exam 1
- Sep 30: §18 limsup and liminf
- Sep 28: §18 limsup and liminf
- Sep 26: §16 Cauchy sequnces; §18 limsup and liminf (started)
- Sep 23: §16 Monotone sequences, Bolzano-Weierstrass for sequences.
- Sep 21: §14 Convergent sequences; §15 Subsequences and combinations, examples.
- Sep 19: §12 Connected open sets in Rp (finished); §14 Convergent sequences (started)
- Sep 16: §12 Connected sets; Connected sets in R; Connected open sets in Rp (started)
- Sep 14: §11 Compactness and Heine-Borel theorem (finished), corollaries; §12 Connected sets (started)
- Sep 12: §11 Compactness and Heine-Borel theorem;
- Sep 9: §10 Cluster points, Nested Cells and Bolzano-Weierstrass
- Sep 7: §9 Open and closed sets, interior, exterior, boundary points; §10 Cluster points (started)
- Sep 2: §8 Vector spaces, inner products, norms, distance, §9 Open Sets
- Aug 31: §3 Finite and Countable sets, §8 Vector spaces, inner products, norms (started)
- Aug 29: §7 Nested Intervals, Cantor Set
- Aug 26: §6 Completeness property of R (continued)
- Aug 24: §5 Order properties of R; §6 Completeness property of R (started)
- Aug 22: §4 Algebraic properties of R; §5 Order properties of R (started)

## Tuesday, November 8, 2011

### Midterm Exam 2

Scheduled Thur, Nov 10, 8-9:30pm, in LYNN G167

We will have Review on Wed, Nov 9. The exam will cover §§19-25, 27-28

[Practice Problems]

Classes to be cancelled to compensate for both midterm exams:
- Nov 4, Nov 16, Nov 21.

## Wednesday, September 28, 2011

### Midterm Exam 1

Scheduled: Tue, Oct 4, 8:00-9:30pm, in LYNN G167

The exam will cover §§3-12, 14-16, 18 of [B].

We will have a Review on Mon, Oct 3.

[Practice Problems]

## Wednesday, August 3, 2011

### Course Information

Schedule: MWF 2:30-3:20pm in REC 302

Instructor: Arshak Petrosyan
Office Hours: MWF 12:30-1:30pm, or by appointment, in MATH 610

Course Description: Basic real analysis, limits, continuity, differentiation, and integration.

Prerequisite: MA 17400 or MA 18200 or MA 26100 or MA 26300 or MA 27100

Textbook:
[B] R. Bartle, The Elements of Real Analysis, Second Edition, John Wiley & Sons, New York, 1975.

Course Outline:
• The algebraic, ordering, and completeness properties of the real numbers. (3 hrs.)
• Topology of Rp. (5 hrs.)
• Sequences in Rp. Convergence and Uniform convergence. Lim Sup and Lim Inf. (5 hrs.)
• Continuous and uniformly continuous functions. sequences of continuous functions. Approximation Theorems. (7 hrs.)
• Differentiation. Mean Value and Taylors Theorem. (3 hrs.)
• The Riemann (Riemann-Stieltjes) Integral. Improper Integrals. (6 hrs.)
• Infinite Series of constant and functions. Absolute and Uniform Convergence. Weierstrass M-Test; Dirichlet and Abel Test. Power Series. Double Series and the Cauchy Product. (8 hrs.)
• Selected Applications of Basic Material (8 hrs.)
• a) Fourier Series
• b) Stone-Weierstrass Theorem
• c) Existence and Uniqueness Theory of Ordinary Differential Equations
(Time should permit to do two of the above applications).
Homework will be collected weekly on Wednesdays. The assignments will be posted on this website at least one week prior the due date. Exams: There well be two midterm exams (evening exams) and a comprehensive final exam (covering all topics). The exact time and place will be specified as the time approaches.