Instructor |
Dragos Oprea
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Lectures: |
MWF 1:00-1:50PM, PETER 102 |
Course Assistants |
Daniel Drimbe
- Discussion 1: APM 7421,
Wednesday, 6:00-6:50PM
- Discussion 2: APM 7421, Wednesday 7:00-7:50PM
- Office: APM 6414
- Office
hour: Monday 10-11, Wednesday 10-11AM, Thursday 12-1.
- Email:
ddrimbe at
ucsd dot edu.
|
Course Content |
First course in a rigorous, entirely proof-based, three-quarter
sequence on real
analysis. Topics include: the real number system, basic point-set topology,
numerical sequences and series, continuity. Students
may not receive credit for both Math 140A and Math 142A. The two
courses cover similar topics, but Math 140A does so in more
depth and from a more theoretical perspective.
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Prerequisities: | Math 31CH (Honors Vector Calculus)
or
Math 109 (Mathematical Reasoning), or consent of the instructor. Math 140
is a difficult and time consuming course, so enroll only if your course
load allows it. |
Grade Breakdown | The grade is computed as the
following
weighed average:
- Homework 20%, Midterm I 20%, Midterm II
20%,
Final Exam 40%.
|
Textbook: | W. Rudin, Principles of Mathematical
Analysis, Third Edition. |
Readings | Reading the sections of the textbook
corresponding to
the assigned homework exercises is considered part of the homework
assignment. You are responsible for material in the assigned reading
whether or not it is discussed in the lecture. It will be expected that
you read the assigned material in advance of each lecture. |
Homework |
Homework problems will be assigned on the
course
homework
page. There will be 7 problem sets, due Fridays at
4:30PM
in
the TA's mailbox. You may work together with your classmates
on your
homework
and/or ask the TA (or myself) for
help on assigned homework problems. However, the work you turn in must be
your own. No late homework assignments will be accepted.
|
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Midterm Exams | There will be two midterm exams given
on October 24 and November 21. There will be no makeup
exams.
Regrading policy: graded exams will be handed back in
section.
Regrading is not possible after the exam leaves
the room.
|
Final Exam | The final examination will be held on
Monday, December 15, 11:30-2:30AM. There is no
make up final
examination. It is your responsability
to ensure that you do not have a schedule conflict during the final
examination; you should not enroll in this class if you cannot
sit for the final examination at its scheduled time. |
Announcements and
Dates |
- Friday, October 3: First lecture
- Friday, October 24: Midterm I
- Tuesday, November 11: Veterans' Day.
- Friday, November 21: Midterm II
- Thursday-Friday, November 27-28:
Thanksgiving. No class.
- Friday, December 12: Last Lecture
- Monday, December 15: FINAL EXAM,
11:30-2:30.
|
Exams |
- Preparation for Midterm 1:
- Preparation for Midterm 2:
- Preparation for Final Exam:
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Lecture Summaries | -
Lecture 1: Introduction and motivation. Ordered sets. Lower
bound, upper bound. Infimum and
supremum.
- Lecture 2: Fields and their axioms. Examples. Ordered fields.
The
definition of the field of real numbers.
- Lecture 3: Comments on the construction of the real
numbers. Archimedian property. Density of rational numbers. Existence of
nth roots of positive real numbers.
- Lecture 4: The extended real number system. Complex numbers.
Definition and properties.
- Lecture 5: Cauchy-Schwartz inequality. Euclidean spaces and
properties. The triangle inequality. Sets, functions, bijections.
- Lecture 6: Countable and uncountable sets. Rational numbers
are countable. Real numbers are uncountable. Countable union of countable
sets is countable. Finite products of countable sets are countable.
- Lecture 7: Metric spaces and examples. Neighborhoods.
Interior points. Open sets and properties.
- Lecture 8: Limit points. Isolated points. Closure. Closed
sets. Dense sets. Properties of closed sets. Supremum belongs to the
closure. Open/closed sets of metric subspaces.
- Lecture 9: Open covers. Compact sets. Compactness is not a
relative concept. Compact sets are closed. Closed sets of compact sets are
compact. Nonemptyness of intersections of compact sets.
- Lecture 10: Nested closed and bounded intervals have non-empty
intersections. Infinite subsets of compact sets have limit points.
Generalizations to k-cells.
- Lecture 11: k-cells are compact. In R^k, compact sets are
closed and bounded and conversely. Sequential compactness and
Weierstrass' theorem.
- Lecture 12: Convergence in metric spaces and its properties.
Limits of real sequences and their behaviour with respect to addition and
multiplication.
- Lecture 13: For monotonic sequences, boundedness and
convergence are equivalent. Sequantial compactness. Cauchy sequences and
convergence. Complete metric spaces.
- Lecture 14: Diagram connecting the type of sequences we
encountered. Special sequences and their limits. Limits at infinity.
Liminf and limsup.
- Lecture 15: Series and their sums. Cauchy criterion. Comparison
test. Series with non-negative coefficients.
- Lecture 16: Geometric series. Cauchy condensation test.
Harmonic series.
- Lecture 17: Two definitions of the number e in terms of
limits of series/sequences. Irrationality of
e.
- Lecture 18: Root test and ratio test. Comparison of the
two tests. Some examples.
- Lecture 19: Power series and radius of convergence. Abel
summation. Alternating series test.
- Lecture 20: Absolute convergence. Addition and multiplication
of power series. Cauchy product of series. Mertens theorem on products of
series.
- Lecture 21: Rearrangements of series. Absolute convergence and
convergence of rearranged series. Limits of functions.
- Lecture 22: Limits of functions and connection with limits of
sequences. Continuity. Properties of continuous functions: sums, products, composition, etc.
- Lecture 23: Continuity and topology: preimages of open and
closed sets are open and closed. Images of compact sets are compact.
Weierstrass extreme value theorem. Discussion of the continuity of
inverses.
- Lecture 24: Uniformly continuous functions. Lipschitz
continuity implies uniform continuity. Continuous functions over compact
sets are uniformly continuous.
- Lecture 25: Connected sets. Intervals of the real line are
connected and
conversely. Images of connected sets under continuous maps are connected.
Intermediate value property. Darboux functions.
- Lecture 26: One-sided limits. Discontinuities
of first and
second kind. Examples. Discontinuities of Darboux functions are of second
kind.
Discontinuities of monotonic functions are of first kind.
- Lecture 27: Monotonic functions only have at most countably
many discontinuities. Limits at infinity and to infinity.
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