Page 95 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
1 1
n q n p
|f (x)|  |y | q |x | p
i i
i 1 i 1
Using the definition of norm for f, we get

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

|y | q
x = i , y 0 and x = 0 if y = 0 … (3)
y i
i i i i
1 1
n p n |y | q p p
Then, x = |x | p i … (4)
i
i
i 1 i 1 |y |
Since q = p (q – 1) we have from (4),

1
n q
x = |y | q … (5)
i
i 1
n n q
|y |
Now f (x) = x y i y i y i … (4)
i
i 1 i 1 i
n
= |y | q (By (3))
i
i 1
So that

n
q
|y | = |f (x) f x … (6)
i
i 1
From (5) and (6) we get,
1
n 1 p
|y | q f
i
i 1

1
n q
|y | q f … (7)
i
i 1
Also from (2) and (7) we have

1
n q
f = |y | q , so that
i
i 1

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