TMA4230 Functional analysis 2005

Lecturer:

Harald HancheOlsen

Literature:

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

Assorted notes on functional analysis (updated 2005–05–11).

On the uniform convexity of L^{p}

Links:

A brief English–Norwegian dictionary covering some much used terms

A different proof of the Uniform boundedness theorem. Best for onscreen 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 final exam will be written (see also Messages, below).
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 N⊆X is a closed subspace. Show that the canonical map Q:X→X/N (defined by Q(x)=x+N) is open.
Assume furthermore that T:X→Y is bounded, and N⊆ker T. Show that there is a unique linear map R:X/N→Y 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 onetoone 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=T^{2}.

Messages (most recent first)
