Page 222 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 18: The Natural Imbedding of N in N**

To show that N* is also of dimension m, we have to prove that there is a uniquely determined Notes
basis (f , f , …, f ) in N*, with f (x ) = .
1 2 m j i y
By the above fact, for each i = 1, 2, …, m, a unique f in N* exists such that f (x ) = . We show now
j j i ij
that {f , f , …, f } is a basis in N* to complete our proof.
1 2 n
Let us consider f + f + …… f = 0 … (1)
1 1 2 2 m m
For all x N, we have f (x) + f (x) + …… + f (x) = 0.
1 1 2 2 m m
We have j j f (x ) 0 j i j i for i = 1, 2, …, m, when x = x . i
j
j j
f , f , …… f are linearly independent in N*.
1 2 m
Now let f (x ) = .
i i
Therefore if x = x , we get
i i
d
f (x) = f (x ) + f (x ) + …… + f (x ) … (2)
1 1 2 2 m m
Further f (x) = f (x ) + …… + f (x ) + …… + f (x )
j 1 j 1 i j i m j m
f (x) =
j j
From (1) and (2), it follows that
f (x) = f (x) + f (x) + …… + f (x)
1 1 2 2 m m
= ( f + f + …… + f ) (x)
1 2 2 m m
(f , f , …… f ) spans the space.
1 2 m
N* is m-dimensional.

18.2 Summary

 The map J : N N** defined by
J (x) = F x N,
x
is called the natural imbedding of N into N**.
 If the map J : N N** defined by

J (x) = F x N,
x
is onto also, then N (or J) is said to be reflexive. In this case we write N = N**, i.e., if N = N**,
then N is reflexive.
 Let N be an arbitrary normal linear space. Then each vector x in N induces a functional F
x
on N* defined by F (f) = f (x) for every f N* such that F = x .
x x
18.3 Keywords

Natural Imbedding of N into N**: The map J : N N** defined by
J (x) = F x N,
x
is called the natural imbedding of N into N**.

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