Final MSc. Exam
Department of MathematicsFerdowsiUniversity of Mashhad
1. State carefully the following theorems :
i. Radon-Nikodym theorem.
ii. Riesz representation theorem for , Hilbert space.
2. Show that every finite dimensional normed space is banach.
3. Let X be a locally compact Hausdorff space and . show that :
A is bounded if and only if for each is bounded . ( Hint : Use the uniform boundedness theorem.)
4. Let X and Y be banach space and . Show that :
and is closed in Y if and only if there exists such that , for all .
5. Show that everynon-zero Hilbert space is isometrically isomorphic to for some non-empty set I .Under what condition I is countable?