Chapter 8 Bounded Linear Operators on a Hilbert Space
188 Bounded Linear Operators on a Hilbert Space (a) If P : X ! X is a projection, then X = ranP kerP. (b) If X = M N, where M and N are linear subpaces of X, then there is a projection P : X ! X with ranP = M and kerP = N. Proof. To prove (a), we rst show that x 2 ranP if and only if x = Px. If x = Px, then clearly x 2 ranP.
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