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¡Þ, with the usual norm ||{x(n)}|| = sup {|x(n)| : n=1, 2, 3,...} is a Banach space.

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

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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<¡Þ .

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 ¡Ê H : ||y|| ¡Ü 1}. Show that if a vector x in H with ||x||=1 can be written as x=¦Ëy+(1?¦Ë)z for y, z ¡Ê B1 and 0<¦Ë<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 ¡Ê B(H) : ||T|| ¡Ü 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)?