HW7_2011 copy - b Show that I is a bounded operator that is...

Math 602 Homework 7
Please note the following very useful inequalities:
| f ( t ) | ≤ k f k
∞
for all t ∈ S if k f k
∞
= sup
x ∈ S
| f ( x ) |
and
Z
b
a
f ( t ) dt ≤
Z
b
a
f ( t ) dt for a < b
1. Consider the sequence of functions f
n
( x ) = x n .
a) Show that the sequence f
n
is point-wise convergent for all x ∈ [0 , 1],
that is, show that for all x ∈ [0 , 1] the limit lim
n →∞
f
n
( x ) = f ( x ) exists; what
is the limit function f ( x )?
b) The functions f
n
belong to the space C [0 , 1] of continuous functions
on [0 , 1]. Do they converge in the sup norm in this space?
c) The functions f
n
belong to the space L
2
[0 , 1]. Do they converge in the
L
2
norm in this space?
2. Consider the integral operator I : C [0 , 2] → C [0 , 2] given by
If ( x ) =
Z
x
0
f ( t ) dt
a) Show that I is a linear transformation.