Page 197 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes For x  , we have defined
n
p
1
n p
p
x = x i
i 1
n n
Now |f (x)| = x y i x y i
i
i
i 1 i 1
By using Hölder’s inequality, we get
1 1
n n p n q
p q
x y x y
i i i i
i 1 i 1 i 1
so that
1 1
n q n p
q p
|f (x)| y i x i
i 1 i 1
Using the definition of norm for f, we get

1
n q
q
f y i … (2)
i 1
Consider the vector, defined by

q
y i
x = , y 0 and x = 0 if y = 0 … (3)
i y i i i
i
Then

1
1 p
n p n q p
p y i
x = x i … (4)
i 1 i 1 y i
Since q = p (q – 1) we have from (4),
1
n p
q
x = y i … (5)
i 1
Now

n n q
y i
|f (x)| = x y i y i
i
y
i 1 i 1 i
n
q
= y i , (By (3))
i 1

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