Page 333 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Now let x belongs to the range of some P so that P x = x. Then
i i
2 2
x P x
i
n
2 2 2
Px P x ... P x
1
j
n
j 1
n
P x,x
j [Using (i)]
j 1
P x,x ... P x,x
1 n
P 1 P ... P x
2
n
Px,x
Px 2 [by (1)]

x 2 ...(2)
Thus we conclude that sign of equality must hold throughout the above computation. Therefore
we have

n
2 2
P x P x
i
j
j 1
2
Px O if j i
j
P x O, j i
j
Px O, j i
j
x is in the null space of P ,i j
j
x M ,if j i
j
x is orthogonal to the range M of every P with j i.
j
j
Thus every vector x in the range P (i = 1,...,n) is orthogonal to the range of every P with j i.
i j
Therefore the range of P is orthogonal to the range of every P with j i. Hence
i j
PP j O, i j [By theorem (8)]
i
Finally in order to show that P is the projection on M M M ... M
1 2 n
We are to show that R(P) = M where R(P) is the range of P.
Let x M. Then x x x ... x
1 2 n
2
where x M , 1 i n. Now from (2), we observe that x 2 Px if x is the range of some P .
i i i
x M, i.e. the range of P .
i i

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