Prove that if M is a dense linear subspace of a separable Hilbert space H, then...

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Prove that if M is a dense linear subspace of a separable Hilbert space H, then H has an orthonormal basis consisting of elements in M. Does the same result hold for arbitrary dense subsets of H? Prove that if M is a dense linear subspace of a separable Hilbert space H, then H has an orthonormal basis consisting of elements in M. Does the same result hold for arbitrary dense subsets of H?