Complex Hilbert Space Definition

This is a definition, so it is not proved from more fundamental propositions within this node.

Let $\mathcal{H}$ be a vector space whose scalars belong to the complex numbers $\mathbb{C}$.

An inner product is a function $\langle \cdot,\cdot\rangle$ that assigns a complex number $\langle\phi,\psi\rangle$ to each ordered pair of vectors $\phi,\psi\in\mathcal{H}$.

The inner product induces a norm on each vector $\psi\in\mathcal{H}$:

A sequence of vectors $(\psi_n)$ is a Cauchy sequence when the distance $\|\psi_n-\psi_m\|$ can be made arbitrarily small by choosing sufficiently large $n$ and $m$.

The space $\mathcal{H}$ is complete when every Cauchy sequence of vectors in $\mathcal{H}$ converges to a vector that is also contained in $\mathcal{H}$.

A complex vector space equipped with an inner product and satisfying this completeness condition is, by definition, a complex Hilbert space.