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A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.

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Sufficient criteria for $X \subset \mathcal{H}$ to be a Lipschitz (or unif. cont.) retract o...

I think what you are looking for is Kirszbraun theorem: https://en.wikipedia.org/wiki/Kirszbraun_theorem
Rafa E.'s user avatar