Real Analysis Final MSc. Exam 1386-10-25 Dr H.R.E.Vishki
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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?
