2004 Indian Statistical Institute M.Sc Mathematics Functional Analysis University Question paper for exam preparation. Question paper for 2004 Indian Statistical Institute M.Sc Mathematics Functional Analysis University Question paper, Exam Question papers 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2012 university in india question papers. SiteMap
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2004 Indian Statistical Institute M.Sc Mathematics Functional Analysis University Question paper

University Question Papers
2004 Indian Statistical Institute M.Sc Mathematics Functional Analysis University Question paper
Time: 3 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 2 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 2 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 2 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]
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