Week 1: 1. Introduction and motivations, finding the maximum and minimum of a continuous function on a bounded closed interval, Riemann integrability; 2. The real number system, the least upper bound property; 3. The insufficiency of the rational numbers.

Homework #1 (due 9/30)

Week 2: Note (uncountability of R). 1. Construction of R, cuts; 2. Completeness of R or the least upper bound property; 3. Archimedean property; 4. Denseness of Q in R; 5. Decimal representations; 6. Countability and uncountability; 7. Metric and metric spaces.

Homework #2 (due 10/7), 3.5, 3.6, 3.7, 3.12, 3.16, 3.28, 3.32, 3.33, 3.46, 3.48.

Week 3: 1. Open and closed sets; 2. Topology and topological spaces; 3. Relative open sets; 4. Component intervals; 5. Representation of an open set in R; 6. Adherent and accumulation points; 7. The closure of a set; 8. Boundary of a set; 9. The Bolzano-Weierstrass theorem; 10. The Cantor intersection theorem; 11. Covering and open covering.

Homework #3 (due 10/14), 3.21, 3.23, 3.25, 3.39, 3.42, 4.2, 4.4, 4.12, 4.18, 4.19.

Week 4: 1. Heine-Borel theorem; 2. Compact sets; 3. Relation between a compact and a close, bounded set; 4. Convergence of a sequence; 5. Cauchy sequences; 6. Complete metric spaces; 7. Limit of a function; 8. Continuous functions.

Homework #4 (不用交), 4.24, 4.25, 4.27, 4.34, 4.41, 4.42, 4.43, 4.54, 4.55, 4.56.

Week 5: 1. Characterization of continuity by the inverse images of open or closed sets; 2. Invariant of compact sets under continuous mappings; 3. Homeomorphic and homeomorphism; 4. Connectedness and pathwise connectedness; 5. Invariant of connected sets under continuous mappings; 6. Connected components; 7. Uniform continuity; 8. Contraction mappings.

Homework #5 (due 10/28), 4.63, 4.64, 4.65, 4.70, 4.71, 4.72.

Week 6: 1. Connected components; 2. Fixed-point and contraction mappings; 3. Local existence of an initial-value problem; 4. Discontinuous functions; 5. Characterization of the set of discontinuities of a monotonic function.

Homework #6 (due 11/4), 5.5, 5.9, 5.12, 5.14, 5.18, 5.24, 5.26, 5.29, 5.33.

Week 7: 1. Derivatives; 2. Chain rule; 3. Rolle’s Theorem; 4. ; 5. Mean-Value Theorem and Generalized Mean-Value Theorem; 6. Intermediate-Value Theorem for Derivatives; 7. Taylor’s formula; 8. Partial derivatives; 9. Differentiability; 10. Functions of bounded variation; 11. Total variation; 12. Characterization of functions of bounded variation.

Homework #7 (due 11/13), 6.2, 6.7, 6.10, 7.3, 7.4, 7.5, 7.13, 7.14, 7.15.

Week 8: 1. Absolutely continuous functions; 2. Rectifiable curves; 3. Arclength of a rectifiable curve; 4. Riemann-Stieltjes integrals; 5. Integration by parts; 6. Change of variable; 7. Increasing integrators; 8. Riemann’s condition; 9. Upper and lower integrals.

Homework #8 (due 11/18), 7.8, 7.10, 7.15, 7.20, 7.21, 7.23, 7.24, 7.26, 7,28, 7.29.

Week 9: 1. 1. Step functions as integrators; 2. Functions of BV as integrators; 3. Sufficient and necessary conditions for the existence of Riemann-Stieltjes integrals; 4. Lebesgue’s criterion; 5. Sets of measure zero; 6. Mean Value Theorem for Riemann-Stieltjes integrals; 7. First Fundamental Theorem of Calculus.

Homework #9 (due 11/25), 7.32, 7.34, 7.35, 7.37, 8.1, 8.4, 8.6, 8,8.

Week 10: 1. Second Fundamental Theorem of Calculus; 2. Second Mean Value Theorem and Bonnet’s Theorem; 3. Riemann-Stieltjes Integrals depending on parameters; 4. Differentiation under the integral sign; 5. Exchange the order of integration; 6. Convergence of a sequence; 7. Limit superior and limit inferior.

Homework #10 (due 12/2), 8.15 d),f),h),i),l),n), 8.16, 8.17,8.22, 8.25, 8.26, 8.27, 8.30, 8.31.

Week 11: 1. Series; 2. Inserting and removing parentheses; 3. Alternating series; 4. Absolute and conditional convergences; 5. Rearrangement of a series; 6. Tests for convergence of positive series: comparison, limit comparison, integral test, Ratio test, Root test; 7. Tests for convergence of a series: Dirichlet’s test and Abel’s test; 8. Summation by parts; 9. Subseries; 10. Double sequence and double series.

Homework #11 (due 12/9), 8.32, 8.34, 8.37, 8.38, 8.39, 8.42, 8.44.

Week 12: 1. Multiplication of two series; 2. Cauchy products; 3. Dirichlet products; 4. Cesaro summability; 5. Convergence of an infinite product; 6. Cauchy criterion.

Homework #12 (due 12/25), 9.2, 9.4, 9.5, 9.6, 9.8, 9.9, 9.10, 9.11, 9.14, 9.16, 9.19, 9.21, 9.23, 9.25.

Week 13: 1. Absolute convergence of an infinite product; 2. Euler’s product for the Riemann zeta function; 3. Pointwise convergence of a sequence of functions; 3. Uniform convergence of a sequence of functions; 4. Uniform convergence and continuity; 5. Uniform convergence and integrability; 6. Cauchy’s criterion for the uniform convergence; 7. Infinite series of functions; 8. Weierstrass M-test; 9. Dirichlet and Abel tests ; 10. Uniform convergence and differentiation; 11. Mean convergence.

Homework #13 (due 12/30), 9.27, 9.29, 9.30, 9.32, 9.34.

Week 15: 1. Mean convergence; 2. Power series; 3. Disk of convergence and radius of convergence; 4. Real analytic functions; 5. Taylor’s formula.

Homework #14 (due 1/6，下午5點前交給楊峻明助教), 9.35, 9.36, 9.37, 9.38.

Week 16: 1. Bernstein’s theorem; 2. Binomial series; 3. Abel’s theorem; 4. Tauber’s theorem.

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Week 1: 1. Compactness revisited; 2. Sequential compactness; 3. Ascoli-Arzela Theorem; 4. Equicontinuity; 5. Peano method; 6. Weierstrass appoximation theorem.

Week 2: 1. Multivariable differential calculus; 2. Differentiability; 3. Cauchy-Riemann equations; 4. The chain rule; 5. Mean value theorem; 6. Sufficient condition for differentiability; 5. Higher order partial derivatives; 6. Equality of mixed-derivatives; 7. Taylor’s formula; 8. Inverse function Theorem.

Homework #2 (due 3/10), 12.2, 12.4, 12.6, 12.7, 12.10, 12.14, 12.18, 12.22, 12.23.

Week 3: 1. Proof of Inverse Function Theorem; 2. Application of Inverse Function Theorem; 3. Motivation of Implicit Function Theorem; 4. Proof of Implicit Function Theorem; 5. Applications of Implicit Function Theorem; 6. Optimization problems; 7. Second order derivative test; 8. Maximum, minimum, and saddle points.

Homework #3 (due 3/17), 13.1, 13.2, 13.5, 13.7.

Week 4: 1. Optimization under constraints; 2. Method of Lagrange multiplier; 3. The constant rank theorem; 4. Multiple Riemann integrals; 5. Integrability; 6. Iterated integrals; 7. Fubini’s theorem; 8. Integrals over general regions.

Homework #4 (due 3/24), 13.8, 13.9, 13.10, 13.11, 13.12, 13.15, 13.16.

Week 5: no class. 3/31: 1st exam

Homework #5 (due 4/14), 14.2, 14.5, 14.6, 14.7, 14.8, 14.9, 14.11, 14.12, 14.14.

Week 7: 1. Outer and inner Jordan contents; 2. Jordan measurable sets; 3. Zero content; 4. Multiple Riemann integral under change of variables; 5. Jacobian.

Week 8: 1. Change of variables formula; 2. Measure theory; 3. Outer measure; 4. (Lebesgue) Measurable sets; 5. (Lebesgue) Measure.

Homework #6 (due 4/21)

Week 9: 1. sigma-algebra; 2. Borel sets; 3. Other characterizations of measurability; 4. Caratheodory theorem; 5. Measurable sets under change of variables; 6. Non-measurable sets.

Homework #7 (due 4/28), 1. Using Caratheodory characterization to establish that the Lebesgue measurable set is a sigma-algebra. Problems in book: 3.8, 3.9, 3.13, 3.14, 3.15, 3.17, 3.20.

Week 10: 1. Non-measurable sets; 2. Measurable functions; 3. Borel functions; 4. Simple functions; 5. Approximate a measurable function by a sequence of simple functions; 6. Semicontinuous functions; 7. Littlewood’s three principles; 8. Egorov’s theorem; 9. Lusin’s theorem.

Homework #8 (due 5/5), 3.4, 3.5, 3.21, 4.3, 4.4, 4.5, 4.7, 4.8, 4.9, 4.11.

5/12: 2nd exam

Week 11: 1. Convergence in measure; 2. Lebesgue integrals; 3. Measurability and integrability.

Week 12: 1. Monotone convergence theorem for nonnegative measurable functions; 2. Equivalent definition of integrability; 3. Tchebyshev’s inequality; 4. Fatou’s lemma; 5. Lebesgue’s dominated convergence theorem; 6. Lebesgue integral of a general measurable function; 7. Monotone convergence theorem; 8. Fatou’s lemma.

Homework #8 (due 5/19), 4.18, 5.3, 5.4, 5.5, 5.6, 5.7, 5.10.

Week 13: 1. Riemann and Lebesgue integrals; 2. Iterated integrals; 3. Fubini’s theorem; 4. Tonelli’s theorem; 5. Fourier series and Fourier integrals.

Homework #9 (due 6/2), 5.13, 5.20, 5.21, 6.1, 6.5, 6.6, 6.10, 6.11.

No classes in week 14.

Week 15: 1. Orthogonal or orthonormal system; 2. Bessel’s inequality; 3. Parseval’s formula; 4. The Riesz-Fischer theorem; 5. Convergence of Fourier series; 6. Riemann-Lebesgue lemma; 7. Dirichlet’s kernel; 8. Riemann localization theorem; 9. Dini’s test; 10. Jordan’s test.

Homework #10 (due 6/11), Problems in Apostol: 11.3, 11.5, 11.6, 11.10, 11.15, 11.20, 11.21, 11.22.

Week 16: 1. Gibb’s phenomenon; 2. Cesaro summability; 3. Fejer’s theorem; 4. Fourier transform; 5. Convolution; 6. Riemann-Lebesgue lemma; 7. Denseness of continuous functions with compact supports in the space of integrable functions.

Homework #11 (due 6/16), 11.16, 11.23, 11.24, 11.25, 11.26.

Week 17: 1. Fourier inversion theorem; 2. Schwartz class; 3. Fourier-Plancherel transform for square integrable functions.

6/23: Final exam.