Programme (not preliminary anymore)

30-3 Chapter 1

2-4 2 hours exercise class
(1.5: 1 4 5 8 9 10 11 12 14 15 18 19 20 22 23 24 25)

followed by
one hour course (Chapter 1, continued). Home work
1 4 9 10 15 18 24 25, hand in 6-4.

6-4 Exercise class: 1.5:
2 13 17 plus what was left from 2-4.

Course: Chapter 1, continued.
Homework: 2.5 3 4 5 7, hand in 16-4.

9-4 Goede Vrijdag

13-4 No exercise class but three hours course. Chapter 2 (finally).

16-4
One hour exercise class and home work: 2.5: 9 13 14 16 22

hand in 20-4. Course: Chapter 2 continued.

Hint for 22: look at (2.7,2.8,2.9,2.4) and use this
for the Riesz representation y of f

in the closure of
M as well as for the Riesz representation z of F in X.

20-4 Exercise class: 18 19 20 21 (no extra home work).

Exercise 20 is wrong. Replace by: if in the Hahn-Banach Theorem the
linear functional f
has f(x)

bounded above by p(x) and bounded below by -p(-x) on M,

then it extends
to a linear F defined on V satisfying the bound in the exercise.

You only have to do the "induction" step. Note that it suffices to

consider alpha=1 and alpha=-1 in the proof of Hahn-Banach.

In fact Bart observed that the lower estimate follows from
the upper estimate.

Use this new Hahn-Banach in Exercise 21 with M=(c).

Course: Chapter 2 continued.

23-4 Exercise class + homework for 27-4: 18 19 (=M should be less or equal) 20 21. Course: Chapter 2, 3 (begin)

27-4 Chapter 3 continued, no exercise class, actually did sections 3.4, 3.1, 3.2.

30-4 Koninginnedag

4-5 3 hours exercise class. 3.8: 1 2 3 4 5 6 10 11 (NB
operator = linear operator) 12
(bad formulation:

show there is a bijection between both spaces which
is linear and bounded in both directions)

15 16 (any -> some)
19 (do this one under the assumption that N(A)={0},

so that
R(A) is closed <=> A has a bounded inverse. Assuming the inverse to
be unbounded,

there is a sequence in D(A) with norm 1, such
the images go to zero.

Either this sequence has a convergent subsequence,
or it has not.

In both cases you can get a contradiction)
21 22.

Homework: 8 11 19 (with N(A)={0}) 21 22, hand in 11-5.

not done yet: 3.8 4 5 6 10 11 12 15 16

7-5 JB:
Chapter 3 conclusion, course starts 14.45, no exercise class.

Section 3.3 + Thm 2.10. Sections 3.5,3.6,3.7.

11-5 Two hours exercise class 3.8 4 5 6 10 11 12 15 16. One hour course Sections 3.5,3.6,3.7 concluded.

14-5 Not from the book: discussion of spectral
theorem for compact symmetric operators

on a real Hilbert space and
Hilbert space methods for elliptic PDE's.