Advanced questions

Question 1

1. State and prove Hölder’s and Minkowski’s inequalities.

Question 2

2. State and prove the closest point property in a Hilbert space.

Question 3

3. Give examples of different ON-bases for L^2([0, 1]).

Question 4

4. State and prove Parseval’s formula for a given ON-basis in a Hilbert space.

Question 5

5. State and prove Weierstrass’ approximation theorem.

Answer 5

No. Too complicated. But want them numbered, so throwing it in anyways.

Question 6

6. State and prove the Hahn-Banach theorem on Hilbert spaces.

Question 7

7. State and prove the Banach-Steinhaus theorem.

Question 8

8. State and prove Poincaré’s inequality

Question 9

9. Prove that H^k(R^n) (→ C_b(R^n) is a continuous imbedding if s > n/2

Question 10

10. Prove that a densely defined bounded linear operator between Banach spaces X and Y can be uniquely extended to a bounded linear operator defined on all of X.

Question 11

11. Prove that the spectrum of a bounded linear operator is a compact subset of C.

Question 12

12. Prove that for a densely defined selfadjoint operator, the spectrum is contained in R.