[1] Carothers, N.N., “Real Analysis.” Cambridge Univ. Press 2000.

[2] Kolmogorov, A.N., and Fomin, S.V., “Introductory Real Analysis”. Dover, 1970.

[3] Haaser, N.B., and Sullivan, J.A., “Real Analysis.” Dover, 1991.

[4] Rudin, W., “Real and Complex Analysis”, 3d ed. McGraw-Hills, 1987.

[5] Folland, G.B., Real Analysis. Wiley, 1984.

[6] Reed, M. and Simon, B., Methods of Modern Mathematical Physics. 1. Functional Analysis. AP 1972.

[7] Oxtoby, J.C., Measure and Category. Springer-Verlag, 1971.

There will be weekly homework assignments, three quizzes (15+15+10=40%),

and cumulative final exam. The final exam has two parts: in-class part (30%) and take-home part (30%).

Bonuses are also available (up to 10%):

- for class activity (mostly, for the solutions of Hw problems reported in class);

- for bonus problems.

1. Discrete Mathematics, Set Theory, Topology and Calculus Review: functions; equivalence relations and quotient spaces; countable and uncountable sets; cardinality; orders and Zorn lemma; supremum and infimum; limits and limit points; continuity; Bolzano-Weierstrass theorem; uniform continuity of functions on closed intervals; uniform convergence; Cantor discontinuum.

[Carothers] # 1, 2, 10.

[Kolmogorov] # 1.1-1.4, 2.1-2.6, 3.1-3.8.

[Haaser] # 1.1-1.6, 2.3-2.5, 4.1, 4.5.

Hw1 (due 09/07)

Hw1-solutions

Lecture Notes 1

2. Cauchy sequences. Metric spaces. Isometry. Complete spaces. l^p spaces.

[Carothers] # 3, 4, 5, 6.

[Kolmogorov] # 5.1, 5.2, 6.1-6.6, 9.1-9.7, 15.1, 15.2.

[Haaser] 4.2-4.4, 4.11, 7.1-7.4.

Hw2 (due 09/14) Hw2-solutions

Hw3 (due 09/21) Hw3-solutions

3. Completeness. Nested balls. Completion. Baire theorem. Fixed points and contraction mapping theorem.

[Carothers] #7.

[Kolmogorov] # 7.1-7.4, 8.1-8.3.

[Haaser] # 4.6, 4.7, 5.1-5.5.

Hw4 (due 09/28) Hw4-solutions

4. Compactness: open cover; total boundedness; compact sets in finite dimensional spaces; uniform continuity; equivalent metrics and norms and the equivalence of norms in any finite dimensional spaces; equicontinuity and Arzela theorem; compactness of topological spaces.

[Carothers] # 8, 11 (p. 178-182)

[Kolmogorov] # 10.1-10.4, 11.1-11.4, 12.1-12.2.

[Haaser] sections 4.8-4.10.

Hw5 (due 10/05) _ Hw5-solutions

5. Category. The sets of the I and II categories. Baire category theorem.

[Carothers] # 9, 11 (p. 183)

Review I

Colloquium questions

6. Borel $\sigma$-algebra. Lebesgue measure. Outer measure. Riemann Integrability. Measurable sets and their structure.

[Carothers] # 16.

[Kolmogorv] # 4.1-4.4, 25.1-25.2, 26.1-26.2, 27.

Lecture Notes 7

Lecture Notes 8

Hw8 (due 11/02) _

7. Measurable functions. Their sequences. Pointwise convergence. Simple functions and the approximation of measurable functions. Lusin theorem. Egorov theorem

[Carothers] # 17

[Kolmogorov] # 28.1-28.5

8. Lebesgue integral. Monotone Convergence theorem. Fatou lemma. Lebesgue Dominated Convergence theorem.

[Carothers] # 18.

[Kolmogorov] # 29.1-29.2, 30.1-30.3.

9. Convergence in measure. Lp spaces, their completeness.

[Carothers] # 19

[Kolmogorov] # 37.1-37.2.

10. Product measures. Fubini theorem.

[Kolmogorov] 35.1-35.3

11. Borel measures: Riesz representation theorem for positive linear functionals, regularity properties, complete measures.

[Kolmogorov] # 28