Let  be given by

Show that f is locally invertible at all points in .

Find and classify the extreme values of 

Subject to the constraint 

Find the Lebesgue integral of the function f given by

Obtain the second Taylor’s series expansion for the function given by

 Does Cantor’s intersection theorem hold for the metric space  with the standard metric? Justify your answer.

 Consider  given by

Find f(2, 0, -1).

 Find the interior, closure, the set of limit points and the boundary of the set

in  with the standard metric.

Find the interior, closure, the set of limit points and the boundary of the set

in  with the standard metric.

1. State whether the following statements are true or false. Justify your answers. 
a) The outer measure m^ of the set  is 0.
b) A finite subset of a metric space is totally bounded.
c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.
d) The surface given by the equation  can also be described by an equation of the form  in a neighbourhood of the point (0, 0).
e) A real valued function f on [a, b] is continuous if it is integrable on [a, b].

Prove the following result:

 Prove the following result:
Suppose A is a non-zero compact self-adjoint operator on a Hilbert space H over K. Prove that there exists a finite set  of a non-zero real numbers with  and an orthonormal set  in H such that

Further, mention in which step of the proof it is used that A is a compact self-adjoint operator. Explain why?

Prove the following result:

Give an example of a positive operator on 

Give an example of a compact linear map on l².

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

 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 .

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

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

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). 

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

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.