While trying to define an inner product, I realized that what I ended up with was a special case of the following. It seems like it should be a standard way to define an inner product, but I'm not sure what it's called.
Let $V$ be a vector space over $\mathbb{C}$ and let $\varphi: V \rightarrow \mathbb{C}$ be a linear functional. Define $\langle \cdot, \cdot \rangle_{\varphi} : V \times V \rightarrow \mathbb{C}$ as follows for $\alpha$, $\beta$ $\in V$:
$\langle \alpha, \beta \rangle_{\varphi}=\varphi(\alpha)\overline{\varphi(\beta)}$