Page 307 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Let A S. Then
Ax,x Ax,x x H

Ax,x Ax,x x H
By definition A A.

' ' on S is reflexive.
' ' is transitive.

Let A 1 A and A 2 A then
2
3
A x,x A x,x x H.
1 2
and A x,x A x,x x H.
2 3
From these we get
A x,x A x,x x H.
1
3
and A x,x A x,x x H.
2 3
From these we get

A x,x A x,x x H.
3
1
Therefore by definition A A and so the relation is transitive.
1 3
' ' is anti-symmetric.

Let A A and A A then to show that A A .
1 2 2 1 1 2
We have A A A x,x A x,x x H.
1 2 1 2
Also A 2 A 1 A x,x A x,x x H.
2
1
From these we get
A x,x A x,x x H.
1
2
A x A x,x 0 x H.
2
1
A A x,x 0 x H.
1 2
A 1 A 2 0

A A
1 2
' ' on anti-symmetric.
Hence ' ' is a partial order relation on S.

Now we shall prove the next part of the theorem.
(a) We have A A A x,x A x,x x H.
1 2 1 2

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