Real Analysis- Dr Sh.Hejazian
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Exam Name : Real Analysis
MSC Exam
Department of Mathematics Ferdowsi University of Mashhad
Second Term of 1386-87
Dr Sh.Hejazian
1. State each of the following :
Signed measure, positive set, negative set, null set.
Give an example of two measures 
and 
s.t. 
.
Give an example of two measures and s.t. 
.
2. State the Lebesgue Radon Nikodym theorem.( no proof !)
3.
(a) let X be a normed space and 
. show that there exists 
s.t. 
and 
.
(b) let 
be linearly independent. Show that there exists s.t. 
and 
.
4.
(a) Let X and Y be normed spaces, define a closed linear operator
.
(b) state and prove the closed Graph Theorem.
(c) give an example of a closed operator which is not bounded.
5.Let H be a Hilbert space.
(a) What is an orthonormal set in H?
(b) What is a basis in H?
(c) Mis a closed linear subspace of H , show that 
.
6.
(a) Define 
for 
.
(b) Let 
be defined by
What is 
?
7. Let X be a finite dimensional linear space. Show that all norms on X are equivalent.