Page 293 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
x, T * y 1 T * y .
2
T* is linear.

T* is bounded
for any y H, let us consider

2
T * y T * y,T * y
TT * yy

TT * y y using Schwarz inequality

T T * y y

2
Hence T * y T T * y y 0 ...(1)
If T * y 0 then T * y T y because T y 0

Hence let T * y 0.

Then we get from (1)

T * y T y .

since T is bounded,

T M so that

T * y M y for every y H.

T* is bounded.
T* is continuous.
Uniqueness of T*.

Let if T* is not unique, let T’ be another mapping of H into H with property
Tx,y = x,T * y x,y H.

Then we have

Tx,y = x,T'y ...(2)

and Tx,y = x,T * y ...(3)

From (2) and (3) it follows that

x,T'y = x,T * y x,y H

x, T'y T * y =0

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