Complex Vector Space with Dot Product is Hilbert Space

Theorem
Let $\C^d$ be a complex vector space with $d$ dimensions.

Let $\innerprod \cdot \cdot$ be the dot product on $\C^d$.

Then $\C^d$ endowed with $\innerprod \cdot \cdot$ is a Hilbert space.

Proof
Let us explore the inner product norm of $\innerprod \cdot \cdot$:


 * $\ds \norm z = \sqrt{ \sum_{i \mathop = 1}^d x_i^2 + y_i^2 }$

and subsequently the metric induced by $\norm \cdot$:


 * $\ds \map d {z, z'} = \norm {z - z'} = \sqrt{ \sum_{i \mathop = 1}^d \paren{ x_i - x'_i }^2 + \paren{ y_i - y'_i }^2 }$