Old Exam Papers June 2012 (Set 2)

6. (a) If x, y be two orthogonal vectors in a Hilbert space
H, then prove that :
2 2 2 2
xy x y x y x y
(b) If le1, e2 ,......, enq be a finite orthonormal set in a
Hilbert space H, and x be any vector in H, then :
n
bx , eig x
2 n
2
and x bx , eige ie jj
i1
i 1
7. (a) Define self-adjoint operator. If T is an operator on a
Hilbert space H, then prove that T is self-adjoint if
and only if Tx x , b g is real for all x. 8
(b) If T1 , T2
are normal operators on a Hilbert space H
with the property that either commutes with the adjoint
of the other, then T1 T2
and T1T2 are also
normal. 8
8. (a) If P be the prefection on a closed linear subspace M
of a Hilbert space H, then prove that M reduces an
operator T if and only if TP PT . 8
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10
bounded set of numbers i.e., lTiq is bounded as
a subset of BbB , Ng.
10
4. (a) If N be a normed linear space and x 0 is a non-zero
vector in N, then a continuous linear functional F
defined on the conjugate space N* such that :
bg
F x x and F
0 0 1
(b) If M be a closed linear subspace of a normed linear
space N and T be a natural mapping (homomorphism)
of N onto N/M such that T x x M
8
bg , then show
that T is continuous (or bounded) linear transformation
with T 1 . 8
5. (a) Define Hilbert space. In a Hilbert space, prove that
the inner product is jointly continuous i.e.,
x x, y y x , y x, y
n n n n
b g b g 6
(b) If M be a closed linear subspace of a Hilbert space
H, x be a vector not in M and d be the distance from
x to M. Then prove that there exists a unique vector
y0 in M such that x y0 d . 10
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