Page 281 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 26.1.2 Theorems and Solved Examples

Theorem 1: Let y be a fixed vector in a Hilbert space H and let fy be a scalar valued function on
H defined by

fy x x,y x H.

Then fy is a functional in H i.e. fy is a continuous linear functional on H and y fy .
*
Proof: From the definition

fy :H C defined as fy x x,y x H.
We prove that fy is linear and continuous so that it is a functional.

x ,x H and , be any two scalars. Then for any fixed y H,
Let 1 2
fy x 1 x 2 x 1 x ,y
2
x ,y x ,y
1 2
fy x 1 fy x 2

fy is linear.
To show fy is continuous, for any x H

fy x x,y x . y ...(1)
(Schwarz inequality)

Let y M. Then for M > 0

fy x M x so that fy is bounded and hence fy is continuous.
Now let y = 0, y 0 and from the definition fy = 0 so that fy y .

Sup fy x
Further let y 0. Then from (1) we have y .
x
Hence using the definition of the norm of a functional,

we get fy y ...(2)

Further fy sup fy x : x 1 ...(3)

y
Since y 0, is a unit vector.
y
From (3), we get

y
fy fy ...(4)
y

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