Page 325 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Let Px = x. Then X is in the range of P because Px is in the range of P.
Px x x M.
Conversely, let x M. Then to show Px = x.
Let Px = y. Then we must show that y = x.

We have
2
Px y P Px Py P x Py
Px Py  P =P
2
P x y 0
x y is a in null space of P.

x y M .
x y z,z M .

x y z.

Now y Px y is in the range of P.
i.e. y is in M. Thus we have expressed

x y z,y M,z M .
But x is in M. So we can write x = x+0, x M,0 M

But H M M .
Therefore we must have y = x, z = 0
Hence x M Px x.

Now we shall show that Px = x Px x .
If Px = x then obviously Px x .

Conversely, suppose that Px x .
We claim that Px = x. We have

x 2 Px I P x 2 ...(1)

Now Px is in M. Also P is the projection on M.
I P is the projection on M .

I P x in M .

Px and I P x are orthogonal vectors.
Then by Pythagorean theorem, we get

2 2 2
Px I P x Px I P x ...(2)

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