Page 324 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 30: Projections

Thus Pz P x y Notes
Px Py

Px x y N

Pz,z x,z  Pz = P x+y x,P being projection on H

x,x y
x,x x,y

x 2

and Pz*,z z,Pz

x y,x x,x x,y

2
x .
Hence Pz,z Pz*,z z H

P P * z,z 0 z H

P P* 0 i.e. P P *
P is self adjoint.

Further, M N N M

If N M , then N is a proper closed linear subspace of the Hilbert space M and therefore a
vector z 0 0 M s.t. z 0 N.
Now z M and z N and H = M N.
0 0
z H z 0, a contradiction.
0 0
Hence N M
Conversely, let P* P,x,y be any vectors in M and N respectively. Then

x,y Px,y

x,P * y x,Py
x,0 0

M N.
This completes the proof of the theorem.
Theorem 2: If P is the projection on the closed linear subspace M of H, then

x M Px x Px x .
Proof: We have, P is a projection on H with range M then, to show x M Px x.

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