MTH 401: Theory of Computation
Department of Mathematics and Statistics
Indian Institute of Technology - Kanpur
August 31, 2016
Time: 10 minutes
Maximum Score: 10
Quiz 2
Name
Roll Number
Consider the following non-deterministic machine over the alphabet c

MTH 401: Theory of Computation
Department of Mathematics and Statistics
Indian Institute of Technology - Kanpur
August 17, 2016
Time: 10 minutes
Maximum Score: 10
Quiz
Name
Roll Number
Let = cfw_0, 1. Draw a deterministic finite state machine, M = (Q, , ,

FUNCTIONAL ANALYSIS: NOTES AND PROBLEMS
Abstract. These are the notes prepared for the course MTH 405 to
be offered to graduate students at IIT Kanpur.
Contents
1. Basic Inequalities
2. Normed Linear Spaces: Examples
3. Normed Linear Spaces: Elementary Pr

Assingment 3 (Hints)
(1) For X a nls. and x 6= y X, show that there is a bounded linear functional f on X such that
f (x) 6= f (y).
Hint: Since x y 6= 0, use Hahn Banach Extension theorem to get a non-zero functional such that
f (x y) 6= 0.
(2) Suppose W

Assingment 1
(1) For 1 p < r < , show that `p `r . Also show that the inclusion is proper.
(2) Show that for 1 p < , c00 ( `p ( c0 ( c ( ` . That is, show the inclusions are proper. Here
c00 denotes the set of all finitely supported sequences, and c is th

Quiz (Hints)
MTH-405A
(1) Why `p with |x|p :=
X
|xi |p
!
is a normed linear space for 1 p < ? [5 points]
i=1
(2) Give an example of a normed linear space X with two norms on it which are not equivalent. Justify
your answer. [5 points]
Done in class.
(3) S

Assignment 5
(1) Let X be an inner product space.
cfw_< xn , yn > is a Cauchy sequence.
If cfw_xn and cfw_yn are Cauchy sequences in X, prove that
(2) Let cfw_Hn be a sequence of Hilbert spaces. Let H denote the set of all sequences cfw_xn with xn Hn

Assingment 4
(1) Let X and Y be Banach spaces over C and T : X Y be a bounded linear operator. If T is bijective
prove that there are constants K1 and K2 such that
K1 |x| |T x| K2 |x|
for all x X.
(2) Define the operator S : `2 `2 by S(cfw_x1 , x2 , x3 ,

Assingment 4 (Hints)
(1) Let X and Y be Banach spaces over C and T : X Y be a bounded linear operator. If T is bijective
prove that there are constants K1 and K2 such that
K1 |x| |T x| K2 |x|
for all x X.
Hint: Use Open Mapping Theorem
(2) Define the oper

Assingment 3
(1) For X a nls. and x 6= y X, show that there is a bounded linear functional f on X such that
f (x) 6= f (y).
(2) Suppose W a subspace of X such that its closure is not X, that is, W is not dense in X. Using
Hahn Banach separation theorem, s

Assingment 2
(1) Denote C 1 [0, 1] the normed linear space of all continuously differentiable functions with respect
to the sup-norm |.|
q. Show that this is not a Banach space with respect to this norm. Consider
the sequence fn (t) =
t2 + n1 . Show that