Complex Hilbert Space

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $\lVert\cdot\rVert$ be the induced norm, and let $d$ be the function assigning to each pair $u,v$ of elements of $V$ the real number $d(u,v)=\lVert u-v\rVert$, which is a metric on $V$ by claim 3 of The Induced Norm is a Norm, and Induces a Metric.

The space $V$ together with $\langle\cdot,\cdot\rangle$ is a complex Hilbert space if the metric space $(V,d)$ is complete.