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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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A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra

One example of such a bimodule $\mathcal E$ is $B$ itself, with the inner product $\langle b,b'\rangle_B = b^* b'$. Choose $B\supset B_0 = C(X)$ a unital abelian $*$-subalgebra, which can be identifi …
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