TMA4225
Analysens grunnlag
– Foundations of Analysis
Fall 2008
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Messages
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- The
lecture on Friday 08.15-10 in F3 has been moved to Thursday 14.15-16
in F4.
- The
originally scheduled fifth hour for exercises has been cancelled and
will be incorporated in the lectures.
- The Midterm Exam will be held on
Thursday October 2, 14.15 – 16, in F4. It will cover the
material up to and including the Bounded Convergence Theorem (Theorem
1.4 on page 56). Midterm
solutions.
- The last
topic of the course will be differentiation in the context of Lebesgue integrals. The lectures will be based on
pages 97-113 in H. L. Royden “Real
Analysis”, Third Edition, 1988. The pages have been copied and
will be handed out in class on Tuesday, November 11. They can also be
downloaded here.
- Last
year’s exam can be downloaded here.
- A note
on limsup, liminf
and accumulation points is available here.
(November 24: The original note
was replaced by a revised version; in particular: an example was
added.) It also contains the solution to Problem 3 in Homework 5.
- A final
meeting before the exam is scheduled for Thursday Nov. 27, 14.15-16,
in F4.
- Dec. 3: Here
are the solutions to the final exam.
Dec. 16: A couple of misprints
have been corrected, and a footnote has been added, in the solution to
Problem 3.
- Dec. 16:
Final grades will be sent to the NTNU Registrar tomorrow (Dec. 17). The
distribution of grades is as follows:
A: 5 B: 1 C: 4 E: 1 F: 5
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Lecturer:
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Trond Digernes
Room 1056, Sentralbygg II.
Telephone 73 59 35 17.
Email: digernes@math.ntnu.no
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Lectures:
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Tuesday 08.15 - 10, F2
Thursday 14.15 – 16, F4
Material covered:
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Week 34-37
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Sections 1.1-1.3, Subsection 1.4.1
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Week 38
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Tuesday Sep 16
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Subsection 1.4.2
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Thursday Sep 18
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Subsection 1.4.3 + exercises
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Week 39
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Tuesday Sep 23
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Section 2.1, p. 49-55
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Thursday Sep 25
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Exercises
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Week 40
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Tuesday Sep 30
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Exercises + Midterm 2007
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Thursday Oct 2
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Midterm exam.
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Week 41
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Tuesday Oct 7
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Midterm solutions. Riemann integrable
functions.
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Thursday Oct 9
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Lebesgue integral of non-negative
functions, Fatou’s Lemma, Monotone
Convergence Theorem.
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Week 42
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Tuesday Oct 14
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Lebesgue’s Dominated Convergence
Theorem.
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Thursday Oct 16
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P. 67-71: Complex-valued functions, The space of integrable functions, Riesz-Fischer
Theorem. - Exercises.
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Week 43
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Tuesday Oct 21
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P. 71-74: Density Theorem (2.4), Invariance properties, Continuity
of translation
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Thursday Oct 23
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Fubini’s theorem
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Week 44
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Tuesday Oct 28
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P. 80-86: Applications of Fubini’s
Theorem.
Ch. 6: Abstract
Measure and Integration Theory (start)
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Thursday Oct 30
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Ch.6 (continued). Exercises.
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Week 45
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Tuesday Nov 4
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Ch. 6: Carathéodory’s Theorem, The Extension
Theorem.
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Thursday Nov 6
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Conclusion of Ch.
6. Exercises.
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Week 46
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Tuesday Nov 11
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Differentiation (see message above)
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Thursday Nov 13
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Differentiation
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Week 47
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Tuesday Nov 18
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Exercises, including Exam 2007
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Thursday Nov 20
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Week 48
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Thursday Nov 27
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Q&A session before the exam
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Exercises:
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Week 35
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Homework 1
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Section 1.6 (p. 37-38): 1, 2, 3, 4
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Week 36
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Homework 2
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Section 1.6 (p. 39-40): 6, 7, 8, 10, 14
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Week 37-38
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Homework 3
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Section 1.6 (p. 41-44): 11, 15, 16, 19, 20, 21, 22, 26
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Week 39
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Homework 4
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Section 1.6 (p. 44): 28, 29
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Week 40-41
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Midterm
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Week 42
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Homework
5
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Week 44
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Homework 6
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Section 2.5 (p. 90-93): 2, 4, 6, 8, 9, 11, 12, 13, 14, 15, 17
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Week 45
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Homework 7
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Section 6.7 (p. 312): 1, 2, 3
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Week 47
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Homework
8
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Office hour:
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Wednesday 11.15 – 12.
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Textbook:
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Stein & Shakarchi: Real Analysis (Princeton Univ. Press, 2005).
H. L. Royden “Real Analysis” (Third
Edition, 1988): Pages 97-113 (handed out in class).
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Curriculum (final):
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Stein & Shakarchi:
Chapter 1 (except Sect.5)
Chapter 2 (except Sect.4)
Chapter 6: Sect. 1 (except 1.2
Metric exterior measures), Sect. 2.
H. L. Royden “Real Analysis” (Third
Edition, 1988): Pages 97-113.
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Midterm exam:
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Thursday October 2, 14.15 – 16, in F4.
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Final exam:
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Monday, December 1; 4 hours, written.
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