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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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The role of completeness in Hilbert Spaces
There are many uses of completeness. One of the first (and one of the most striking) is the fact that a closed subspace has a complement. This in turn uses the fact that for a point not on the closed …