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

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Advance Math

Problem 4. Let P be the orthogonalprojection associated with a closed subspace S in aHilbert space H, that is P is a linear operatorsuch that

P(f) = f if f ∈ Sand P(f) = 0 if f ∈ S⊥.

(a) Show that P2 = P andP∗ = P.

(b) Conversely, if P is any bounded operator satisfyingP2 = P and P∗ = P,prove that P is the orthogonal projection for some closedsubspace of H.

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

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