Complex Hilbert Spaces

A vector space (like $\mathbb{R}^n$ or $\mathbb{C}^n$) equipped with an inner product is called a Hilbert space.

In general, a Hilbert space requires that the underlying vectorspace is what's known as "complete". However, for physics purposes, we do not need to concern ourselves with this technicality, as most of the spaces we work in are automatically complete. We will instead focus on understanding the inner product structure of the complex vector space $\mathbb{C}^{n}$.

For $\mathbb{C}^n$, the inner product between two vectors $\vec{v}, \vec{w} \in \mathbb{C}^n$ is defined to be:

$$\vec{v} \cdot \vec{w} = v_1^* w_1 + v_2^* w_2 + \cdots +v_n^* w_n$$

where $v_i, w_i \in \mathbb{C}$ for all $i \in \{1, \dots, n\}$.

You will sometimes see the inner product in $\mathbb{C}^{n}$ written as $\langle \vec{v}, \vec{w} \rangle$.

NOTE (Important): We are taking the complex conjugates of the entries of $\vec{v}$, i.e. the first vector in the inner product. This is standard in physics when dealing with Hilbert spaces like $\mathbb{C}^{n}$.

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