I'm trying to find a proof for the following statement, but I'm out of luck.
For $k\ge 0$ the space $\Gamma(\mathbb{P}^n, \mathcal{O}(k))$ is canonically isomorphic to the space $\mathbb{C}[z_0,\dots,z_n]_k$ of all homogeneous polynomials of degree $k$.
I can't figure out why all global sections $s : \mathbb{P} \to \mathcal{O}(k)$ should be a homogeneous polynomials.
I know that $\mathcal{O}(k) \cong \mathcal{O}(1)^{\otimes k}$ and that $\mathcal{O}(1)$ is the dual of the tautological line bundle $\mathcal{O}(-1)$ on $\mathbb{P}^n$, but this ain't enough for me to be able to prove the statement. If someone here knows where I can find a proof or knows how to prove this I would appreciate that.