I'm reading through Huybrecht's section on Vector Bundles. In general, he notes that for a (holomorphic) line bundle $L$ with $s_{1}, \ldots, s_{k}$ globally generating (holomorphic) sections, we can define a hermitian structure on $L$ by,
$$h(t) = \frac{|\psi(t)|^{2}}{\sum |\psi(s_{i})|^{2}},$$
where $t$ is a point in the fiber $L(x)$ and $\psi$ is a local trivialization of $L$ around $x$.
I'm mostly okay with this, however later in the Chapter, he moves to an example of the tautological bundle $\mathcal{O}(1)$ on $\mathbb{P}^{n}$. We consider $z_{0}, \ldots, z_{n}$ as the globally generating sections, such that the hermitian structure takes the form,
$$h(t) = \frac{|\psi(t)|^{2}}{\sum |\psi(z_{i})|^{2}}.$$
Then working locally over a standard open set $U_{0} \subseteq \mathbb{P}^{n}$, he writes the local hermitian structure as a positive, real function
$$h=(1+\sum|w_{i}|^{2})^{-1},$$
where I'm guessing $w_{i}=z_{i}/z_{0}$. I see obviously how similar this is to the general expression for $h(t)$, but how does one rigorously arrive at this scalar function $h$? I think the way you're technically supposed to view $h(t)$ is as an anti-linear isomorphic from $L \to L^{*}$, so I'm not seeing the connection.