# TMA4230 Functional analysis 2005

Lecturer:
Harald Hanche-Olsen
Literature:
Erwin Kreyszig: Introductory functional analysis with applications, ISBN 0471504599.
Assorted notes on functional analysis (updated 2005–05–11).
On the uniform convexity of Lp
A brief English–Norwegian dictionary covering some much used terms
A different proof of the Uniform boundedness theorem. Best for on-screen viewing: A5 size (pdf, ps). Best for printing: A4 size (pdf, ps).
A note on the Weierstrass theorem
Evaluation:
A midterm test counts 20%. The final exam will be on 31 May.
The syllabus («pensumliste») is finally here.
Final exam
As you are no doubt now aware, I ended up giving you the wrong exam set. Well, only one question was wrong but still. Here is the one you got (combined English/Norwegian).
And for the record, the message that I handed out around 11:30.
You can now look at my suggested solution (new).
In case you are curious, or more to the point for future reference, the exam as I had intended it to be: English, Norwegian.
And in case you wonder how I managed the mixup: Very easy. I created two files, one with the English text and one with the Norwegian text. Then I merged the two into one for printing. When I updated the parts, I forgot to recreate the merged version. Ouch!
Exercises
Solution set 1 (pdf, ps): From Kreyszig – section 4.2: 3, 4, 5, 10, and 4.3: 3, 11.
Solution set 2 (pdf, ps): From Kreyszig – section 4.5: 5, 8, 9, 10; 4.7: 7, 8; plus the extra problem (with hint): Prove that a closed subspace of a reflexive space is reflexive.
Solution set 3 (pdf, ps) for these problems:
Warm up with Kreyszig 2.1: 14 (p 57) and 2.3: 14 (p 71).
Assume that X is a normed space and NX is a closed subspace. Show that the canonical map Q:XX/N (defined by Q(x)=x+N) is open.
Assume furthermore that T:XY is bounded, and N⊆ker T. Show that there is a unique linear map R:X/NY so that T=RQ. What is its norm?
Assume furthermore (still!) that N=ker T. Show that T is open if and only if R has a bounded inverse.
Finally, a challenge: Use the closed graph theorem to prove the open mapping theorem. (Hint: Do it first for one-to-one mappings, then use the above results to get the general case.)
Solution set 4 (pdf, ps) for problem set A, for topology (pdf/ps)
Midterm exam: problem (pdf/ps) and suggested solution (pdf/ps)
Solution set 5 (pdf, ps) for problem set B pdf/ps.
(Problems in problem set B were mistakenly called Exercise A.1, etc. This has now been corrected.)
Problem set 6: From Kreyszig – section 7.2: 3,5; 7.4: 4; 7.5: 5, 6; 7.6: 2, 9, 10; and 7.7: 2, 5, 7.
Also: If T is a bounded operator on a Hilbert space, show that ||T*T||=||T2||.
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Harald Hanche-Olsen Oppdatert: 2005–06–09 16:07