Page 286 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 26: The Conjugate Space H*

2 Notes
y y 0
1 2
y y
1 2
is one-to-one.

*
(iii) is onto: Let f H . Then y H such that
f(x) = (x,y)

since f(x) = (x,y) we get
f = f so that y = f = f.
y y
*
Hence for f H , a pre-image y H. Therefore is onto.
(iv) is isometry; let y ,y H, then
1 2
y y f f
1 2 y y
1 2
f y f
1 y
2
But f y f y f y y y 1 y 2 (By theorem (1))
1 2 1 2
Hence y 1 y 2 y 1 y .
2
(v) To show is not linear, let y H and be any scalar. Then ,y f y. Hence for any
x H, we get

f (x) (x, y) (x,y) f (x)
y y
f y f y

y y
is not linear. Such a mapping is called conjugate linear.
This completes the proof of the theorem.

Note: The above correspondence is referred to as natural correspondence between H and
*
H .
Theorem 4: If H is a Hilbert space, then H is also an Hilbert space with the inner product defined
*
by
(f , f ) = (y,x) … (1)
x y
Proof: We shall first verify that (1) satisfies the condition of an inner product.

Let x,y H and , be complex scalars.
(i) We know (see Theorem 3) that
f y f y

f f f .
y y y
Now f f ,f f f ,f ... (2)
x y z x y z

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