Madurai Kamraj University (MKU) 2007 M.Sc Mathematics Functional Analysis - Question Paper

Functional Analysis

6562/KAA    october 2007

Optional FUNCTIONAL ANALYSIS

(For those who joined in July 2003 and after)

Time : Three hours    Maximum : 100 marks

SECTION A (4 x 10 = 40 marks)

Answer any FOUR questions.

1.    If M be a linear subspace of a normed linear space N, and / be a functional defined on M, then prove that f can be extended to a functional f₀ defined on the

whole space N such that || f_Q\\= || f ||.

2.    State and prove the Uniform Boundedness theorem.

3.    If M is closed linear subspace of a Hilbert space H; x is a vector not in M and d is the distance from x to M, then prove that there exists a unique vector y_Q in M

such that |x _<y₀|j=rf .

4.    If N₁ and N₂ are normal operators on H with the

property that either commutes with the adjacent of the other, then prove that N₁ + N₂ and N_lN₂ are normal.

5.    If I is a proper closed two-sided ideal in A, then prove that the quotient algebra All is a Banach Algebra.

6.    If f_Y and f₂ are multiplicative functionals on A with the same null space M, then prove that = f₂.

7.    Explain the following :

(a)    Conjugate space ofX

(b)    Dual basis

(c)    Completion of the n/s X

(d)    Adjoint of F.

8.    For a compact operator A on a Banach space X prove that Z(A -I) and Z(A'-I) are equal where A is the transpose of A.

SECTION B (3 x 20 = 60 marks)

Answer any THREE questions.

All questions cany equal marks.

9.    Let M be a closed linear subspace of a normed linear space N. Show that N!M is also a normed linear space and that if N is a Banach space then so is N/M .

10.    If B and B' are Banach space and if T is a continuous linear transformation B onto B', then prove that the image of each open sphere centred on the origin in B contains an open sphere centred on the origin in B'.    V

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11.    (a) Show that l₂ is an inner product fc*pace.

(b) State and prove the Bessel's inequality.

II n llⁿ

12.    Prove that r(x) = limp where r(x) is the

spectral radius of an element x in the general Banach algebra A.

13.    State and prove the Gelfand-Neumark theorem.

14.    Let 1 < p < oo and + = 1. For y e V {[a, 6]). Let

P Q

/

f_y e (L^p ([a, 6])) be defined by

b

f_y(x)= jxydm,xeL^p ([a,6]).

a

/

Let F:L⁹([a,&]H(k^p([a,&])) be given by

F(y) = f_y> yeL^q[[a,b]]. Then prove that F is a linear

/

isometry of L^q([a, b]) into (l^p ([a, 6])) and also F is onto if and only if 1 < p < .

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Attachment: madurai-kamraj-university--mku--2007-m-sc-mathematics-22087-1.pdf

Earning:   Approval pending.