LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

CV 58

FOURTH SEMESTER – APRIL 2007

MT 4804 – FUNCTIONAL ANALYSIS

 

 

 

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

 

 

 

Answer all questions.

 

01.(a)   Show that every vector space has a Hamel basis.

(OR)

Prove that a subset S of a vector space X is linearly independent Û for every
subset {x₁, x₂, …, x_n­} of S,  åa_ix_i = 0 Þ a_i = 0, for all  i.                  (5)

(b)(i)  Show that every element of X/Y contains exactly one element of z, where Y
and z are complementary subspaces of X.

(ii)  If Z is a subspace of a vector space X of deficiency 0 or 1, show that there is
an f Î X^* such Z = Z(f).                                                                 (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 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.                         (15)

 

 

2. (a) State and prove F-Riesz Lemma.

(OR)

Let X and Y be normed 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.  (5)

(b)  State and prove the Hahn Banach Theorem for a complex normed linear
space.

(OR)

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.

(15)

 

 

 

 

 

3. (a) State and prove Riesz Representation Theorem.

(OR)

Prove that a real Banach space is a Hilbert space iff the parallelogram law
holds in it.                                                                                             (5)

(b) State and prove the Projection Theorem.

(OR)

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.                            (15)

 

4. (a) State and prove Bessel’s inequality.

(OR)

If T is an operator on a Hilbert space X, show that T is a normal Û its real
and imaginary parts commute.                                                          (5)

(b)(i) If T is an operator in a Hilbert space X, then show that

(Tx, x) = 0  Þ T = 0.

(ii)  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.                                                                                     (7 + 8)

(OR)

(iii) State and prove Riesz-Fischer Theorem.                                        (15)

5. (a) Prove that the spectrum of x, , is non-empty.

(OR)

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.                     (5)

(b)   State and prove the Spectral theorem.

(OR)

Let G be a set of regular elements in a Banach algebra A.               (5)

Prove that f : G ® G given by f(x) = x^-1 is a homeomorphism.

 

 

 

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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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