| Time |
Location |
Instructor |
Office Hours |
E-mail |
| MW 14:00-15:15 | 203 Hylan Building | Dan-Andrei Geba | MW 11:15-12:15, 806 Hylan Building | dangeba@math.rochester.edu |
Syllabus: the structure of the set of real numbers; the extended real line; fundamental inequalities; cardinal numbers; monotone real sequences; recurrent real sequences; Toeplitz's theorem; extremal limits for a real sequence; generalizations of convergence tests for real series; rearrangements; real functions (limits, continuity, differentiation, antiderivatives, and integrability); normed spaces; inner product spaces, Hilbert spaces; linear operators.
Prerequisites: MATH 265 or MATH 265H.
No textbook required. We will often refer for background information to "Principles of Mathematical Analysis" by W. Rudin (on reserve at Carlson) and to "Elementary Analysis" by K. Ross (available online in the UR libraries system). On reserve at Carlson and online, you will also find "Problems in mathematical analysis" (vol. 1-3) by W. Kaczor and M. Nowak.
Success in this course will be achieved only through hard work, massive problem solving, and active participation in class discussions. Please take advantage of my office hours.
Homework is usually assigned weekly on Wednesday, starting 1/24, and it is due back the following Thursday by 17:00. There will be 9 assignments from which the best 7 will count toward your grade. Late homework is not accepted.
The homework should be uploaded to Gradescope as a single PDF file.
Both tests are in-class exams. They are scheduled for Wednesday, 3/6, and Monday, 4/29, respectively. The first exam will be based on material covered from the start of the semester all the way to and including the lecture on 2/28. The second exam will be based on material covered from after the spring break all the way to and including the lecture on 4/22.
1. The course average is not based on a curve, nor on previously fixed scales. It will reflect how well the class is doing, and it will be high if everyone is working hard for the homework and is performing well on the exams.
2. Incomplete "I" grades are almost never given. The only justification is a documented serious medical problem or a genuine personal/family emergency. Falling behind in this course or problems with workload on other courses are not acceptable reasons.
3. If you miss one of the exams with a valid excuse (e.g., illness or emergency), you must notify the instructor and provide supporting documentation verifying your excuse as soon as possible. For a valid excuse with supporting documentation, the other exam will count as your make-up test. If you miss both exams, you are in danger of failing the class. In principle, no make-up exams will be offered. If you miss an exam without a valid excuse (and supporting documentation), you will receive a score of 0 on that test.
4. You are responsible for knowing and abiding by the University of Rochester's academic honesty policy. Any violation of academic honesty will be pursued according to the specified procedures. Furthermore, the following Mathematics Department policy also applies to this class:
Any usage whatsoever of online solution sets or paid online resources
(chegg.com or similar) is considered an academic honesty violation and
will be reported to the Board on Academic Honesty. In particular, any
assignment found to contain content which originated from such sources
is subject to a minimum penalty of zero on the assignment and a full letter
grade reduction at the end of the semester (e.g. a B would be reduced to a C).
This applies even if the unauthorized content was obtained through indirect
means (through a friend for instance) and/or the student is seemingly unaware
that the content originated from such sources. If you have any questions about
whether resources are acceptable, please check with your instructor.
5. This course follows the College credit hour policy for four-credit courses. This course meets 3 academic hours per week. Students may also be expected to deepen their understanding of the course material through close examination/evaluation of the readings assigned in the course.
| Week of | Topic | Reading assignment | Homework |
| 1/15 | The structure of the set of real numbers; the extended real line; fundamental inequalities (Cauchy-Schwarz, Bernoulli, inequalities between various means). | Rudin (chapter 1); Ross (chapter 1). | |
| 1/22 | Cardinal numbers (ordering cardinal numbers, Schroder-Bernstein theorem, operations with cardinal numbers, Cantor's theorem); monotone real sequences (existence result on monotone subsequences, monotone sequences approximating e, asymptotics for partial sums of the harmonic series, problem-solving methodology). | Rudin (section in Chapter 2 on finite, countable, and uncountable sets; section in Chapter 3 on upper and lower limits) | Homework 1 (due 2/1) |
| 1/29 | Recurrent real sequences (closed-form expression, monotonicity, convergence issues), Toeplitz's theorem (consequences, Stolz-Cesaro theorem) | Homework 2 (due 2/8) | |
| 2/5 | Extremal limits for a real sequence (equivalent definitions, subsequential limits set, monotonicity issues), real series (summation by identifying telescopic structure, generalizations of convergence tests) | Homework 3 (due 2/15) | |
| 2/12 | Generalizations of convergence tests for real series (Kummer, Raabe-Duhamel, Bertrand, Gauss convergence tests; Abel-Dirichlet convergence criterion), rearrangements (semi-convergent, absolutely convergent, and unconditionally convergent series; Riemann's theorem) | Homework 4 (due 2/22) | |
| 2/19 | Real functions (monotone functions, functions having the intermediate value property, linear functions, functions having a fixed point), limits, continuity, and differentiation (extremal limits for a function, oscillation of a function at a certain point, characterization theorem for continuous functions, discontinuities, Cauchy's functional equation, theorems of Fermat, Rolle, Lagrange, and Cauchy, and their applications) | Homework 5 (due 2/29) | |
| 2/26 | Differentiability, antiderivatives, and integrability (smooth functions with compact support, antiderivatives, functions which admit antiderivatives and applications, equivalence between Riemann integrability and Darboux integrability, sets with Lebesgue/Jordan measure 0, Lebesgue's characterization theorem for integrable functions) | Review for first exam | |
| 3/4 | Review session (3/4), first exam (3/6) | ||
| 3/18 | Normed spaces (sections 2.1 and 2.2 in Rynne-Youngson) | Banach spaces and inner products (sections 2.3 and 3.1 in Rynne-Youngson) | Homework 6 (due 3/28) |
| 3/25 | Banach spaces, inner products, orthogonality, and orthogonal complements (sections 2.3 and 3.1-3.3 in Rynne-Youngson) | Orthonormal bases in infinite dimensions and Fourier series (sections 3.4 and 3.5 in Rynne-Youngson) | Homework 7 (due 4/4) |
| 4/1 | Orthonormal bases in infinite dimensions and Fourier series (sections 3.4 and 3.5 in Rynne-Youngson) | Continuous linear transformations, the norm of a bounded linear operator (sections 4.1 and 4.2 in Rynne-Youngson) | Homework 8 (due 4/18) |
| 4/8 | Continuous linear transformations, the norm of a bounded linear operator (sections 4.1 and 4.2 in Rynne-Youngson) | The space B(X,Y), inverses of operators (sections 4.3 and 4.4 in Rynne-Youngson) | |
| 4/15 | Homework 9 (due 4/25) | ||
| 4/22 | Review session (4/24) | Review for second exam | |
| 4/29 | Second exam (4/29) |