TMA4230 Functional analysis 2006

Lecturer:

Harald Hanche-Olsen

Literature:

Erwin Kreyszig: Introductory functional analysis with applications, ISBN 0471504599.

Assorted notes on functional analysis (updated 2006-05-11).

Links:

A brief English–Norwegian dictionary covering some much used terms

Exercises:

Problem set 1 (pdf).

Problem set 2 (with apologies for late delivery): From Kreyszig – section 4.2: 3, 4, 5; 4.3: 3, 11; 4.5: 5, 8, 9, 10; plus the extra problem (with hint): Prove that a closed subspace of a reflexive space is reflexive.

Midterm exam and suggested solution; results.

Problem set 3 (pdf).

Evaluation:

A written midterm test counts 20%. The final exam, on May 26, will be oral and counts 80%.

A detailed syllabus (pensumliste) is available.

The oral examination will be as follows: First the student presents a mini lecture (20 minutes) on an assigned topic, assigned from the list of topics below. Then questions from any parts of the syllabus are asked. Total time: 40–45 minutes per student. Because there are many students and the schedule is tight, it is important that you stay within your 20 minutes. You can bring and consult notes for your mini lecture. But all notes have to be put away for the questions. Try to present the main result(s) with some proofs or bits of proof. Obviously, you cannot present everything, but include what you think is most interesting or important. Topics are:

• Uniform boundedness theorem
• Open mapping theorem
• Closed graph theorem
• Hahn–Banach theorem (basic version)
• Hanh–Banach theorem (geometric / separation version)
• Uniform convexity and reflexivity
• Functional calculus (for continuous functions, bounded selfadjoint operator)
• Spectral theorem for bounded selfadjoint operators
• Some sequence spaces and their duals
• L^p spaces

The place, times and assigned topics for the oral examination will be as follows. (Names now removed in the interest of privacy, but I keep the list of topics for later reference.)

Start	Room	Name	Topic
09:00	E204	—	Hahn–Banach theorem (basic version)
09:45	E204	—	Lp spaces
10:30	E204	—	Spectral theorem for bounded selfadjoint operators
11:15	E204	—	Hanh–Banach theorem (geometric / separation version)
13:00	S2–734	—	Open mapping theorem
13:45	S2–734	—	Uniform convexity and reflexivity
14:30	S2–734	—	Uniform boundedness theorem
15:15	S2–734	—	Closed graph theorem
16:00	S2–734	—	Functional calculus (for continuous functions, bounded selfadjoint operator)

Room E204 is in Block E, Electro building; just off the long bridge between buildings on the 2^nd level.
Room 734 is in “Sentralbygg 2” next to the mathematics department office.
Please make sure to appear at the appointed time! And let me know as soon as possible if you are unable to come.

Yet more details concerning the upcoming exam may still appear here.

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Harald Hanche-Olsen Oppdatert: 2006–06–01 17:26