Problem 4. Let P be the orthogonal projection associated with a closed subspace S in a...

Problem 4. Let P be the orthogonal projection associated with a closed subspace S in a Hilbert space H, that is P is a linear operator such that

P(f) = f if f ∈ S and P(f) = 0 if f ∈ S^⊥.

(a) Show that P² = P and P^∗ = P.

(b) Conversely, if P is any bounded operator satisfying P² = P and P^∗ = P, prove that P is the orthogonal projection for some closed subspace of H.

(c) Using P prove that if S is a closed subspace of a separable Hilbert space, then S is also a separable Hilbert space.

Expert Solution

Colby Messinger answered 1 year ago

Let W be a subspace of R^n, and P the orthogonal projection onto W. Then Ker...

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