Hilbert space

Nearest point of a convex subset of a Hilbert space

Let 𝑋 be a Hilbert space and let 𝐴 ⊆𝑋 be a inhabited, closed, convex subset. Then for any 𝑥 ∈𝑋 there exists a unique 𝑎 ∈𝐴 such that ‖𝑥 −𝑎‖ =𝑑(𝑥,𝐴) fun where

𝑑(𝑥,𝐴)=inf{‖𝑥−𝑎‖:𝑎∈𝐴}

Proof

proof

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