The Complex Coordinate Space is a Complex Hilbert Space

Let $n$ be a natural number, let $\mathbb{C}^{n}$ be the complex coordinate space with the standard inner product $\langle\cdot,\cdot\rangle$, which is a complex inner product space by The Standard Inner Product Makes the Complex Coordinate Space an Inner Product Space, let $\lVert\cdot\rVert$ be the induced norm, and let $d$ be the metric $d(u,v)=\lVert u-v\rVert$ of claim 3 of The Induced Norm is a Norm, and Induces a Metric. Then the following hold.

1. (Completeness) The metric space $(\mathbb{C}^{n},d)$ is complete.

2. (Hilbert space) Consequently $\mathbb{C}^{n}$ together with the standard inner product is a complex Hilbert space. In particular, taking $n=2$, the qubit state space is a complex Hilbert space.