Page 202 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 16: Continuous Linear Transformations

Notes
q 1 q 1
Now = sgn sgn
k k k k k k k k
= | | = | | p (Using property of sgn function) … (7)
q
k k k k
1
n p
p
x = k
k 1
1
n q
q
= k … (8)
k 1
Since we can write
n
x = k e k , we get
k 1

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

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

1
n q
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.
q
k
We next show that T is onto  .
q
*
Let ( )  , we shall show that there is a g  such that T maps g into ( ).
q
k p k

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