Page 287 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
But f f ,fz z, x, y (by (1))
x y
Now z, x y z,x z,y

f ,f z f ,f z … (3)
x
y
From (2) and (3) it follows that f x f ,f z f ,f z f ,f z
y
y
x
(ii) f ,f y,x x,y f ,f .
x y x y
(iii) f ,f x,x x 2 f 2 so f ,f 0 and f 0 f 0.
x x x x x x x
i iii implies that (1) represents an inner product. Now the Hilbert space H is a complete
normed linear space. Hence its conjugate space H is a Banach space with respect to the norm
*
defined on H . Since the norm on H is induced by the inner product, H is a Hilbert space with
*
*
*
the inner product (f , f ) = (y,x)
x y
This completes the proof of the theorem.
*
**
Cor. The conjugate space H of H is a Hilbert space with the inner product defined as follows:
*
**
If f,g H ,let F and F g be the corresponding elements of H obtained by the Riesz representation
f
theorem.
**
Then (F ,F ) = (g,f) defines the inner product of H .
f g
Theorem 5: Every Hilbert space is reflexive.
**
Proof: We are to show that the natural imbedding on H and H is an isometric isomorphism.
*
Let x be any fixed element of H. Let F be a scalar valued function defined on H by F (f) = f(x) for
x x
**
*
every f H . We have already shown in the unit of Banach spaces that F x H . Thus each vector
x H gives rise to a functional F in H . F is called a functional on H induced by the vector x.
*
**
x x
**
Let J : H H be defined by J x F for every x H.
x
**
We have also shown in chapter of Banach spaces that J is an isometric isomorphism of H into H .
We shall show that J maps H onto H .
**
*
Let T : H into H defined by
1
T x f ,f y y,x for every y H.
1 x x
**
and T : H * into H defined by
2
*
T f x F ,F f f,f for f H .
2
x
f x
f x
Then T .T is a composition of T and T from H to H . By Theorem 3, T ,T are one-to-one and
**
2 1 2 1 1 2
onto.
Hence T .T is same as the natural imbedding J.
2 1
For this we show that J(x) = (T .T )x for every x H.
2 1
Now (T .T )x = T (T (x)) = T (f ) = F .
2 1 2 1 2 x f x
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