I have this problem which I want to do before I go back to uni. The context was not covered in class before the break, but I want to get my head around the problem before we resume classes. So any help on this is greatly appreciated.(adsbygoogle = window.adsbygoogle || []).push({});

Question

Suppose [itex]C[/itex] is a nonempty closed convex set in a Hilbert Space [itex]H[/itex].

(i) Prove that there exists a unique point [itex]c_0 \in C[/itex] of smallest norm, and that we then have [itex]\Re \langle c_0 \, | \, c-c_0\rangle \geq 0[/itex] for all [itex]c \in C[/itex].

(ii) For any point [itex]x_0 \in H[/itex] show that there is a unique closest point [itex]c_0[/itex] of [itex]C[/itex] to [itex]x_0[/itex], and that it satisfies thevariational inequality[itex]\Re \langle x_0 - c_0 \,|\, c -c_0\rangle \leq 0[/itex] for all [itex]c \in C[/itex]

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# Homework Help: Smallest Norm in a Hilbert Space

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