Undervisningsplan

Solutions will be discussed during the lecture on September 9. 

Exercises 2.10, 3.15, 3.19, 3.21, 3.22, and the following three exercises

Problem 1. Show that the space A of exercise 3.15 is a closed subspace of the Hilbert space l-two, l^2.

Problem 2. Prove that l-infinity, l∞, with the usual norm ||{x(n)}|| = sup {|x(n)| : n=1, 2, 3,...} is a Banach space.

Problem 3. Prove that l∞ with the metric associated to the usual sup- norm is nonseparable.

Section 4.3. The Space B(X,Y). 

The last part of Section 5.1 Section 5.3 (The Hahn-Banach extension theorem in normed spaces, Thm. 5.19, without proof.) 

Problem 1. Prove that the Banach space l^p is reflexive for 1<p<∞ .

Exercises that will not be worked through in detail on the board: 6.1, 6.3, 6.7 (only for the exercise in 6.1), 6.10, 6.13, 6.15

The spectrum of an operator. (Orthogonal projections. Functions of self-adjoint operators.) 

(1) Let H be a Hilbert space and let B1 = {y ∈ H : ||y|| ≤ 1}. Show that if a vector x in H with ||x||=1 can be written as x=λy+(1−λ)z for y, z ∈ B1 and 0<λ<1, then we must have x=y=z. (In this case we say that x is an extremal point in B1). (Hint: remember when there can be equality in the triangle inequality.) (2) Use (1) to show that every isometry in B(H) is an extremal point in the closed unit ball B(H)1 = {T ∈ B(H) : ||T|| ≤ 1} of B(H).

Theorem 6.58 (Positive Square Root), Theorem 6.59 (The Polar Decomposition).

Compact operators (Chapter 7). 

(Examples 4.7 and 4.41)