I try to solve the following questions:
Give an elementary proof of Riemann-Roch theorem for Riemann sphere $X=\hat{\mathbb{C}}$.
and
Let $X$ be a torus, and $p\in X$ a point. Show that $$\dim O(np)=\left\{\begin{array}{ll}0& n<0\\1&n=0\\n&n\geq 1\end{array}\right.$$ without using Riemann-Roch theorem.
I combined the two questions because they look similar. If necessary, I will split them to separate questions.
My attempt: The version of Riemann-Roch theorem I learnt is $$\dim O(D)-\dim O(K-D)=\deg (D)+1-g$$ where $g$ is the genus of Riemann surface, $D$ is a divisor and $O(D)$ is given by the subspace of meromorphic functions on the riemann surface $$O(D)=\{f~|~(f)+D\geq 0\}.$$
In the first exercise, $g=0$. Then it is enough to show that $$\dim O(D)-\dim O(K-D)=\deg (D)+1$$ And pick a canonical divisor: on the chart containing $0$, the meromorphic form is $$z~\mathrm{d}z,$$ on the chart containing $\infty$, the form is $$-\frac{\mathrm{d}~w}{w^3}$$ Then we obtain a canonical divisor $$1\cdot 0-3\cdot\infty$$ Also, I know the number of zeros of a meromorphic function on $\hat{\mathbb{C}}$ is equal to the number of its poles. Then I have no idea how to continue.
For the second one, find the canonical divisor on a torus is much more difficult for me.
For both of problems, to avoid the Riemann-Roch theorem, I try to construct the basis of the above vector spaces to verify the equation of dimension. But I don't know how to do it.
Thanks for any hints and answers!