Page 199 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
y
x = i when |y | = max y i
i y i i 1 i n
and x = 0 otherwise.
i
From the definition, x = 0 k i. So that we have
k
y
x = i = 1
y

n
Further |f (x)| = (x y ) i
i = |y |
i
i 1
Hence |y | = |f (x)| f x
i
|y | f or max. {|y |} [ ||x|| = 1]
i i
||f|| … (10)
From (8) and (10), we obtain
||f|| = max. {|y |} so that
i
y f is an isometric isomorphism of L to  n 1 * .

Hence  n *  n .
1

n
(iii) Let L =  with the norm
x = max {|x | : i = 1, 2, 3, …, n}.
i
Now f defined in (1) above is continuous as in (1).
n
Let L represents the set of all continuous linear functionals on  so that
L =  n * .

Now we determine the norm of y’s which makes y f as isometric isomorphism

n n
|f (x)| = x y i x y i .
i
i
i 1 i 1
n n
But x y i max( x ) y i
i
i
i 1 i 1
Hence we have
n
| f (x) | y i x so that
i 1
n
f y i … (11)
i 1

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