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Problem 6. For a closed convex nonempty subset K of a Hilbert space H and x...

Problem 6. For a closed convex nonempty subset K of a Hilbert space H and x ∈ H, denote by P x ∈ K a unique closest point to x among points in K, i.e. P x ∈ K such that

||P x − x|| ≤ ||y − x||, for all y ∈ K.

First show that such point P x exists and unique. Next prove that all x, y ∈ H

||P x − P y|| ≤ ||x − y||

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