Page 221 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Hence F = F … (7)
x x
Using definition of J and equations (6) and (7) we get
J = F = F + F = J (x) + J (y) … (8)
(x + y) x+y x y
and J = F = F = J (x) … (9)
( x) x x
(8) and (9) J is linear and also (4) shows that J is norm preserving.
For any x and y in N, we have
J (x) – J (y) = F – F = F = x – y … (10)
x y x–y
Thus J preserve distances and it is an isometry. Also (10) shows that

J (x) – J (y) = 0 J (x – y) = 0 x – y = 0
i.e. J (x) = J (y) x = y so that J is one-one.
Hence J defines an isometric isomorphism of N into N**. This completes the proof of the theorem.

n
Example 1: The space  (1 p < ) are reflexive.
p
Solution: We know that if 1 p < , then
 n p * =  .
n
p
But  n q * =  n p

Hence  n p * * =  n p

n
Similarly we have  n * * =  for p = 1
1
1
n
and  n * * =  for p =
n
So that  spaces are reflexive for 1 p < .
p
Example 2: The space  for 1 < p < are reflexive.
p
Sol: We know that if *  and * 
p p q p

 * *  .
q p
 are reflexive for 1 < p < .
p
A similar result can be seen to hold for L (X).
p

Example 2: If N is a finite dimensional normed linear space of dimension m, then N* also
has dimension m.
Solution: Since N is a finite dimensional normed linear space of dimension m then {x , x , …, x }
1 2 m
is a basis for N, and if ( … ) is any set of scalars, then there exists a functional f on N such
1 2 m
that f (x ) = , i = 1, 2, …m.
i i

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