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

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. 

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

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

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

 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.

) Which of the following maps are open? Give reasons for your answer.
i)  given by .
ii)  given by .

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.

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

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.]

Consider the space C¹[0,1] 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 . 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.

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

State whether the following statements True or False? Justify your answers: 
a) The function  defined on  as:
 for 
is a norm.
b) C₀ is a Banach space.
c) If A is the right shift operator on l², then the eigen spectrum is non-empty.
d) If a normed linear space is reflexive, then so is its dual space.
e) If a normed linear space X is finite dimensional, then so is X'.

) C₀ is a Banach space.

State whether the following statements True or False? Justify your answers: 
a) The function  defined on  as:
 for 
is a norm.

) Using fourth order Taylor series method with , solve the initial value problem

upto .

 Find an approximate value of y(1.0) for the initial value problem 

y = x - 2y,  y(0) = 1

using Milne-Simpson’s method

with the step length . Calculate the starting value using Runge-Kutta fourth order method with the same h.

Using standard five point formula, solve the Laplace equation  in R where R is the square  subject to the boundary conditions  on
 and  on . Assume .

 Find an approximate value of y(1.0) for the initial value problem

using the multiple method

with step length . Calculate the starting values using Runge-Kutta second order method with the same h.

Solve the wave equation  with the initial and boundary conditions

.

with , using the explicit method upto four time levels.