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Real Analysis

Final MSc. Exam

Department of MathematicsFerdowsiUniversity of Mashhad

1386-10-25

Dr H.R.E.Vishki

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?