Page 283 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 2
Noting that y ,y y 0, we get from (3),
0 0 0
f y
f x 0 2 x,y 0 ...(4)
y 0
We can write (4) as

f y
f x x, 0 2 y 0
y
0

f y
Now taking o y as y, we have established that there exists a y such that f(x) = (x,y) for x H.
y o
o
Step 2: In this step we know that

f y
If f =0, then y = 0 and f y hold good.

Hence let f 0. Then y 0.

From the relation f(x) = (x,y) and Schwarz inequality we have
f x x,y x y .

f x
sup y .
x 0 x
Using definition of norm of f, we get from above

f y ...(5)

Now let us take x = y in f(x) = (x,y), we get
2
y y,y f y f y

y f ...(6)
(5) and (6) implies that

f y .

Step 3: We establish the uniqueness of y in f(x) = (x,y). Let us assume that y is not unique in
f(x) = (x,y).

Let for all x H, y ,y such that
1 2
f(x) = (x,y ) = (x,y )
1 2
Then (x,y ) – (x,y ) = 0
1 2
(x,y – y ) = 0 x H.
1 2

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