# Show that a space of holomorphic sections of a line bundle is isomorphic to the space of meromorphic functions on a Riemann surface.

Let $$\Sigma$$ be a Riemann surface, $$x \in \Sigma$$ and let $$z: U \rightarrow D \subset \mathbb{C}$$ be a local coordinate system centered at $$x$$. For every $$k \in \mathbb{Z}$$, a holomorphic line bundle $$L_{kx}$$ on $$\Sigma$$ is defined by gluing the trivial vector bundles $$(\Sigma \backslash \{x\}) \times \mathbb{C} \rightarrow \Sigma$$ and $$U \times \mathbb{C} \rightarrow U$$ via the holomorphic transition function $$(U \backslash \{x\}) \times \mathbb{C} \rightarrow (U \backslash \{x\}) \times \mathbb{C}, \ (p, v) \mapsto (p, z(p)^k v)$$

Can someone give me some hints how to prove the following claims ?

The space of global holomorphic sections of $$L_{kx}$$ is isomorphic to the space of meromorphic functions on $$\Sigma$$, holomorphic outside of $$x$$ and has a pole of order at most $$k$$ if $$k \geq 0$$ and a zero of order $$k$$ if $$k < 0$$.