LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 48

FOURTH SEMESTER – APRIL 2008

MT 4804 – FUNCTIONAL ANALYSIS

 

 

 

Date : 16/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer ALL questions:                                                                    (5 x 20 = 100)

 

1. a)         (i) Show that every element of X/Y contains exactly one element of Z

where Y and Z are complementary subspaces of a vector space X.

(OR)

(ii) Prove that every vector space has a Hamel Basis. (5)

 

1. (i) Prove that a subset S of a vector space X is linearly independent for every subset of S, for all i.

(ii) If prove that the null space has deficiency 0 or 1 in X.

Conversely, if Z is a subspace of X of deficiency 0 or 1, show that there is an such that .                                                 (7+8)

(OR)

(iii) Let X be a real vector space, let Y be a subspace of X and p be a real valued function on X such that and p(ax)=a p(x) x,yX, for . If is a linear functional on Y and    prove that there is a linear function F on X such that and .                                  (15)

 

1. a)         Let X and Y be normal linear spaces and let T be a linear transformation

of X into Y. Prove that T is bounded if and only if T is continuous.

(OR)

State and prove F-Rierz Lemma.                                                     (5)

 

1. State and prove Hahn Banach Theorem for a complex normal linear space.

(OR)

State and prove the Uniform Boundedness Theorem. Give an example to

show that the Uniform Boundedness Principle is not true for every normal

vector space.                                                                                   (15)

1. a)         Prove that a real Banach space is a Hilbert space iff the parallelogram law

holds in it.                                                                                        (5)

(OR)

Let X and Y be Banach spaces and let T be a linear transformation of X

into Y. Prove that if the graph of T is closed, then T is bounded.

1. State and prove Projection Theorem.

(OR)

State and prove Open Mapping Theorem.                                      (15)

 

 

1. a)         If T is an operator on a Hilbert Space X, show that T is normal its real

and imaginary parts commute

(OR)

If T is an operator on a Hilbert space X, prove that (                                                              (5)

1. (i) If N₁ and N₂ are normal operators on a Hilbert space X with the

property that either commute with adjoint of the other, prove that N₁+N₂ and N₁N₂ are normal.

(ii) If M and N are closed linear subspaces of a Hilbert space X and if P and Q are projections on M and N, then show that                                                        (8+7)

(OR)

(iii) State and prove Rierz – Fischer Theorem.

1. a)         Prove that the spectrum of  is non-emply.

(OR)

Show that given  is continuous.                    (5)

1. State and prove the Spectral Theorem.

(OR)

Define spectral radius and derive a formula for the same.               (15)

 

 

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