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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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separable spaces-QM vs. functional analysis

In Nouredine Zettili's QM book, Hilbert space $H$ is said to be separable when: There exists a Cauchy sequence $\psi_n \in H$ ($n=1,2,\ldots)$ such that for every $\psi$ of $H$ and $\varepsilon > 0,$ …