LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

CV 5

FOURTH SEMESTER – APRIL 2006

                                                      MT 4800 – FUNCTIONAL ANALYSIS

 

 

Date & Time : 20-04-2006/FORENOON     Dept. No.                                                       Max. : 100 Marks

 

 

ANSWER ALL QUESTIONS

1. a) i) Show that every vector space has a Hamel basis

(or)

1. ii) If f Î X^*, prove that the null space Z(f) has deficiency 0 or 1 in X.  Conversely, show that if Z is a subspace of X of deficiency 0 or 1, then there is an f Î X^* such that Z=Z(f).
2. b) iii)   Show that every element of X/Y contains exactly one element of Z, where Y and Z are complementary subspaces of X.
3. iv) Let X and Y be normed linear spaces and let B(X, Y) denote the set of all bounded linear transformations from X into Y.  Then prove that B(X,Y) is a normed linear space.

(or)

1. v) Let X be a real vector space, p be a real valued function on X such that P(x+y) £ p(x) + p(y) and p(ax) = a p(x) ” x,y Î X and a ³ 0, and let Y be a subspace of X.  If f is a linear functional on Y and f(x)  £  p(x) ” x Î Y, prove that there is a linear functional F on X such that F(x)=f(x)  ” x Î Y and F(x) £ p(x) ” x Î
2. a) i) If x ¹ 0 is an element of a real normed linear space X, then show that there exists an x Î x¢ such that x¢(x) = ||x||  and ||x¢|| = 1.

(or)

1. ii) 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.                                                                   (8)

(or)

1. b) iii)   State and prove the uniform boundedness theorem.
2. iv) Give an example to show that uniform boundedness principle is not for every normed vector space.                                                                                                             (10+7)

(or)

1. v) Let X and Y be Banach spaces and if T is a continuous linear transformation of X onto Y, then prove that T is an open mapping.                                                                               (17)
2. a) i) State and prove the Riesz – Representation Theorem.

(or)

1. 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 M ^N Û PQ = 0 Û QP=0                                               (8)
2. b) iii)   If T is an operator on a Hilbert space X, then prove that T is normal iff its real and imaginary parts commute.
3. iv) Prove that how Hilbert spaces are isomorphic iff they have the same dimension.  (7+10)

(or)

1. v) If P is a projection on a closed linear space M of a Hilbert space X, prove that M is invariant under T Û TP =PTP
2. vi) If P₁, P₂, … P_n are projections on closed linear subspaces M₁, M₂, … M_n on X, then prove that P= P₁ + P₂ + …+P_n is a projection iff the P_i are pairwise orthogonal and in the case P is a projection on M=M₁+M₂+…+M_n.                                                                              (5+12)
3. a) i) Prove that every element x in a Banech algebra A for which ||x–1|| < 1, is regular, and the inverse of such an element is given by .

(or)

1. ii) Let A be a Banech algebra and x Π Then prove that the spectrum of x, s(x), is non-empty.   (8)
2. b) iii)   Let G be the set of regular elements in A and S be the set of singular elements in A.  Prove that G is an open set and therefore S is a closed set.
3. iv) Show that the mapping x à x^–1 of G into G is continuous and is therefore a homeomorphism.   (5+12)

(or)

1. v) State and prove the Spectral Theorem.                                                    (17)

 

 

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