What is the $L^2$ space of sections of a vector bundle? The vector bundle in question can be a holomorphic line bundle with a hermitian metric.
Given $\mathcal{L}$ a holomorphic line bundle with transition functions $g_{ij}$ a section is defined as a collection ${h_i}$ such that $h_i=g_{ij}h_j$. Or is another definition of sections better for this purpose?
Oh i should probably mention that this is on a compact Riemann surface.
 A: Let $L \to \Sigma$ be your holomorphic line bundle. This bundle admits, by a partition of unity argument and the local trivialization of the bundle, a smooth bundle metric, lets call it $g_L$. Given a global section $s:\Sigma \to L$, we define
$$
|s|^2=g_{L}(s,s).
$$
This is well defined and we can just integrate it:
$$
||s||_{L^2}^2=\int_{\Sigma}|s|^2 d\mathrm{vol} 
$$
In more detail:
So given a trivialization atlas $\{(U_i,\phi_i) \}$ of $L$, this section looks locally like
$$
s:U_i \to U_i \times  \mathbb{C}, z \mapsto (z,u(z))
$$
and a local metric looks like
$$
g_i:U_i\times \mathbb{C}\times \mathbb{C} \to \mathbb{R},(z,u_1,u_2) \mapsto u_1 \overline{u_2}.
$$
You can glue these local metrics together using a partition of unity argument to build a (global) metric $g_L$.
To see that this is well defined for your section, use the fact that for a hermitian vector bundle, the transition functions can be chose unitary, in your case: $g_{ij} \in U(1)$. But you see that every local metric is invariant under $u(1)$ transformations:
$$
e^{i\phi}u_1 \overline{e^{i\phi}u_2}=u_1 \overline{u_2}
$$
and so is a combination of them: The (global) metric! Now, this gives therefore a well-defined function for a section $s: \Sigma \to L$ in the form
$$
|s|^2: \Sigma \to \mathbb{R}, z \mapsto g_L(s,s) .
$$
This is now something you can integrate over $\Sigma$. Now, every section with finite $L^2$ norm is in $L^2(\Gamma(L))$ (the $L^2$-space of sections of $L$).
A good exercise is to check what this would look like for the space of sections of the trivial bundle $\Sigma \times \mathbb{C} \to \Sigma$ for a (non-)compact Riemann surface. Hint: It is the ordinary $L^2$ norm
$$
\int_{\Sigma} |u(z)|^2 d\mathrm{vol}.
$$
I just added non-compact to allow non-trivial, holomorphic sections.
