Ques 1.

Let 

Prove that Y is a proper subspace of X. Is Y a closed subspace of X? Justify your answer

Ques 2.

Let  and  Find  for P = 4  and .

Ques 3.

Let E be a subset of a normed space X, Y = span E and  Show that  if and only if f(a) = 0 whenever  and f = 0 everywhere on E

Ques 4.

Consider the space c₀₀ For  define  Show that f is a linear functional which is not continuous w.r.t the norm 

Ques 5.

Consider the space  of all C¹ functions on [0,1] endowed with the uniform norm induced from the space C[0,1] and consider the differential operator  defined by Df = f'. Prove that D is linear, with closed graph, but not continuous. Can we conclude from here that C¹[0,1] is not a Banach space? Justify your answer.

Ques 6.

When is a normed linear space called separable? Show that a normed linear space is separable if its dual is separable [You should state all the proposition or theorems or corollaries used for proving the theorem]. Is the converse true? Give justification for your answer. [Whenever an example is given, you should justify that the example satisfies the requirements.]

Ques 7.

Let X be a Banach space, Y be a normed linear space and f be a subset of B(X,Y).If f is not uniformly bounded, then there exists a dense subset D of X such that for every  is not bounded in Y.

Ques 8.

Read the proof of the closed graph theorem carefully and explain where and how we have used the following facts in the proof. 
 i) X is a Banach space.
 ii) Y is a Banach space.
 iii) F is a closed map. 

iv) Which property of continuity is being established to conclude that F is continuous.

Ques 9.

Which of the following maps are open? Give reasons for your answer.

i)  given by T(x,y,z) = (x,z)

ii)  given by T(x,y,z) = (x,y,0)

Ques 10.

Let  be given by  Show that f is continuous w.r.t the supnorm and f is not continuous w.r.t the p-norm.

Ques 11.

Let X be an inner product space and  Prove that  if and only if 

Ques 12.

Let  and F be the set of all  in  such that . Find  Verify that every  can be expressed as  where  and  .

Ques 13.

Given an example of an Hilbert space H and an operator A on H such that  is empty. Justify your choice of example.

Ques 14.

Let A be a normal operator on a Hilbert space X. Show that  where  denotes the approximate eigen spectrum of A and  denotes the spectrum of A.

Ques 15.

Let  with  Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example

Ques 16.

Give one example of each of the following. Also justify your choice of example.

i) A self-adjoint operator on  .

ii) A normal operator on a Hilbert space which is not unitary.

Ques 17.

Let X be a normed space and Y be proper subspace of X. Show that the interior Y⁰ of Y is empty

Ques 18.

Let X,Y be normed spaces and suppose BL(X,Y) and CL(X,Y) denote, respectively, the space of bounded linear operators from X to Y and the space of compact linear operators from X to Y. Show that CL(X,Y) is linear subspace of BL(X,Y). Also, Show that if Y is a Banach space, then CL(X,Y) is a closed subspace of BL(X,Y).

Ques 19.

Define a Hilbert-Schmidt operator on a Hilbert space H and give an example. Is every Hilbert-sehmidt operator a compact operator? Justify your answer.

Ques 20.

Let {A_n} be a sequence of unitary operators in BL(H). Prove that if A    then A is unitary.

Ques 21.

Define the spectral radius of a bounded linear operator .Find the spectral radius of A in ,where A is given by the matrix

with respect to the standard basis of  .

Ques 22.

Let X be a Banach space and Y be a closed subspace of .X Let  be canonical quotient map. Show that  is open.

Ques 23.

State giving reasons, if the following statement are true or false.

a) A closed map on a normed space need not be an open map.

b) c₀₀ is a closed subspace of 

c) The dual of a finite dimensional space is finite dimensional.

d) If T₁ and T₂ are positive operators on a Hilbert space H, then T₁ + T₂ is a positive operator on H.

e) On a normed space X, the norm function  is a linear map.