LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

1. Sc. DEGREE EXAMINATION – MATHEMATICS

IV SEMESTER – APRIL 2004

MT 4800/M 1025 – FUNCTIONAL ANALYSIS

 

Date   :                                                                                                            Max. Marks  :  100 Marks

Hours           :   3 hours

 

Answer All questions:                                                                         

 

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

(Or)

If  , prove that the hull space Z(f) has deficiency O or 1 in X.  Conversely, show that if Z is a subspace of X of deficiency O or 1,   then there is  an   such  that

Z = Z(f).                                                                                                          (8)

 

1. 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. Then prove that B(X, Y) is a normed linear space.

 

(ii) Let X and Y be normed linear spaces and let T : XY be a linear transformation.  Prove that T is bounded if and only if T is continuous. (9 + 8)

 

(Or)

State and prove the Hahn – Banach Theorem (real version)           (  17 )

 

2. a) 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.

(Or)

State and prove F-Riesz  Lemma                                                    (8)

 

1. State and prove the uniform boundedness theorem. Give an example to show that the uniform boundedness principle is not true for every normed vector space.

(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.                                    (17)

 

3. a) State and prove Bessel’s  inequality

(Or)

If T is an operator on X, then show that (Tx, x) = 0

 

1. b) i) If x^‘ is a bound linear functional on a Hilbert space X, prove that there is a unique

such that  x^‘(x) = (x,

 

1. 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    PQ =  O QP = O.      (9 + 8)

(Or)

Prove that two Hilbert spaces are isomorphic iff they have the same dimension. (17)

 

 

4)  a)  Define a topological  divisor of Zero.  Let  S be the set of singular elements in a Banach   algebra .  Prove that the set of all topological divisors of Zero is a subset of S.

 

(Or)

Let A be a Banach algebra  and   Then prove that the spectrum of x, is non-empty.

 

1. State and prove the Spectral Theorem.

(Or)

 

Define the spectral radius of an element x in a Banach Algebra   A.  In the usual notation, prove that .                                                                                      (17)

 

 

 

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             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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    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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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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2012

MT 4810 – FUNCTIONAL ANALYSIS

 

 

Date : 16-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

Answer ALL questions:                                                                                           (5 x 20 = 100 Marks)

1. a) 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)

If , prove that the null space has deficiency 0 or 1 in a vector space X.

Conversely, if Z is a subspace of X of deficiency 0 or 1, show that there is an

such that .                                                                                              (5)

1. b)  Prove that every vector space X has a Hamel basis and all Hamel bases on X have the

same cardinal number.                                                                               (6+9)

(OR)

Let X be a real vector space, let Y be a subspace of X and   be a real valued function

on X such that and  for  If f

is a linear functional on Y and prove that there is a linear

functional F on X such that  and     (15)

2. a) Let X and Y be normed linear spaces and let T be a linear transformation of X onto
3. Prove that T is bounded if and only if T is continuous.

(OR)

If is an element of a normed linear space X, then prove that there exists an

such that  and .                                                           (5)

1. b) State and prove Hahn Banach Theorem for a Complex normed 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 normed vector space.         (9+6)

 

 

 

 

3. a)  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, than T is bounded.

(OR)

If x¹ is a bounded linear functional on a Hilbert space X, prove that there is a unique

such that .                                                         (5)

1. b) If M is a closed subspace of a Hilbert space X, then prove that every x in X has a

unique representation  where .

(OR)

State and prove Open Mapping Theorem.                                                    (15)

4. a) If T is an operator on a Hilbert space X, show that T is normalits real and imaginary parts commute.

(OR)

If and  are normal operators on a Hilbert space X with the property that either

commute with adjoint of the other, prove that and are normal.

1. b) (i) If T is an operator on a Hilbert space X, prove that

(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       (6+9)

(OR)

State and prove Riesz-Fischer Theorem.                                                        (15)

 

 

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

(OR)

Show that given by  is continuous, where G is the set of regular

elements in a Banach Algebra.                                                                        (5)

1. b) State and prove the Spectral Theorem.

(OR)

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

 

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