LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

CV 54

FOURTH SEMESTER – APRIL 2007

MT 4800 – FUNCTIONAL ANALYSIS

 

 

 

Date & Time: 16/04/2007 / 9:00 – 12:00       Dept. No.                                                       Max. : 100 Marks

 

 

Answer ALL questions.

 

01.(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 a subset S of a vector space X is linearly independent  Û for
every subset  {x₁, x₂, …, x_n} of S,  i.       (8)

 

(b)(i)    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.  Show that B(X, Y) is a
normed linear space and B(X, Y) is a Banach space, if Y is a Banach space.

(OR)

(ii)    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 p(x) ³ 0, p(x + y) = p(x) + p(y) and
p(ax) = a(Px) for a ³ 0.  If f is a linear functional on Y such that

£ p(x) for every  x Î Y, show that there is a linear functional F on X
such that F(x) = f(x) on Y and  £ p(x) on X.                       (17)

 

02.(a)(i)    Show that  a normed vector space is finite dimensional iff the closed and
bounded sets are compact.

(OR)

(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)

 

(b)(i)  State and prove the Uniform Boundeness Theorem.  Give an example to
show that the uniform boundedness principle is not true for every normed
vector space.                                                                                   (9 + 8)

(OR)

(ii)  If X and Y are Banach spaces and if T is a continuous linear transformation
of X onto Y, then prove that T is an open mapping.                       (17)

 

03.(a)(i)  Let X be a Hilbert space and S = {x_a} _{a Î A}  be an orthonormal set in X.

Prove that S is a basis iff it is complete in X.

(OR)

(ii)  If T is an operator on a  Hilbert space  X, then show that

(Tx, x) = 0 Þ T = 0.                                                             (8)

 

 

(b)(i)  State and prove Riesz Representation Theorem

(ii)  If M and N are closed linear subspaces of a Hilbert space H and if P and Q
are projections on M and N, then show that M ^ N Û PQ = O Û QP = 0.

(OR)                                                    (9 + 8)

(iii)  State and prove Riesz – Fischer Theorem.                                      (17)

 

04.(a) (i)  Prove that the spectrum of x, s(x), is non-empty.

(OR)

(ii) Define a Banach Algebra A, set of regular elements G, set of singular
elements S, and prove that G is open and S is closed.                   (8)

 

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

(OR)

(ii)   State and prove the Spectral theorem.                                           (17)

 

 

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