Indian Statistical Institute (ISI) 2004 M.Sc Mathematics Functional Analysis - Question Paper

Time: three hrs Date:20-07-04 Max. Marks : 100

1. Let X be a n-dimensional normed linear space. Let L : X ! Cn be a
linear map. Show that L is continuous. [15]

2. Let M = ff two C([0; 1]) : f0 exists and is continuousg. De¯ne
jjfjj¤ = jjfjj + jjf0jj. Show that jjjj¤ is a norm on M. [10]

3. State and prove the open mapping theorem. [15]

4. Let X and Y be Banach spaces. Let T two L(X; Y ) be a compact
operator. Show that T¤ is a compact operator. [15]

5. Let ffngn¸1 ½ L2(R) be a complete ortho normal sequence. De¯ne
ª : L2(R) ! `2 by ª(f) = (R f ¹ fndx)n¸1. Show that ª is an onto
isometry. [15]

6. Let H be a Hilbert space. Suppose N two L(H) is a normal operator.
Show that ¸ is an eigen value of N if and only if ¹¸ is an eigen value of
N¤. [15]

7. Let K be a compact Hausdor® space. Let f : K ! `2 be a continuous
map. De¯ne T : `2 ! C(K) by T(®)(k) =< ®; f(k) >. Show that T is
a well-de¯ned, bounded linear map. [15]
1

Earning:   Approval pending.