Denote the universal bundle of $\mathbb{C}P^1$ by $U$, the dual bundle by $H$. Prove that the Holomorphic tangent bundle over $\mathbb{C}P^1$ is isomorphic to $H\otimes H$, and calculate the dimension of the space of global holomorphic sections(i.e. holomorphic vector fields).
I'm new on complex geometry. There are some questions similar to mine on MSE and I am trying to understand some of them. I wonder is there a direct proof or some reference for my question.
Appreciate any help!