Page 100 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 8: Bounded Linear Functional on the L -spaces
p

Notes
q 1
q 1
Now | | sgn k | | sgn k
k k k k k
= | | = | | p … (7)
q
k k k k
(Using property of sgn function)
1
n p
x = | | p
k
k 1
1
n p
x = | | p
k
k 1
1
n q
= | k | q … (8)
k 1
Since we can write
n
x = e , we get
k k
k 1
n n
f (x) = k f(e ) k k
k
k 1 k 1
n
f (x) = | | q (Using (7)) … (9)
k
k 1
We know that for every x  p

| f (x) | f x ,
which upon using (8) and (9), gives

1
n n p
|f (x)| | k | q f | k | q
k 1 k 1
which yields after simplification.

1
n p
| | q
k f … (10)
k 1
Since the sequence of partial sum on the L.H.S. of (10) is bounded; monotonic increasing, it
converges. Hence

1
n p
| | q
k f … (11)
k 1
So the sequence ( ) which is the image of f under T belongs to  and hence T is well defined.
k q

LOVELY PROFESSIONAL UNIVERSITY 93