Hi I'm reviewing Riemann Surfaces and this is a question from an old text of exam. We did until Riemann-Roch Therorem. I can't understand which space $L$ is, therefore the real meaning of this answer. I hope there's someone who can explain what's the text asking; I'd really appreciate it. Thank you.
Let $X = \mathbb{P}^1(\mathbb{C})$, $\quad $ $p_1 = [0,1] = 0 \quad p_2 = [1,0] = \infty$
(1) Compute the dimension of $L(p_1 + p_2)$.
(2) Write a base for $L(p_1 + p_2)$.
The vector subspace $L(p_1+p_2)\subset Rat (\mathbb P^1)=\mathbb C(z)$ consists of those rational functions which have at worst simple poles at $p_1$ and $p_2$, so that $$L(p_1+p_2)=\mathbb C\oplus \mathbb Cz\oplus \mathbb C \frac 1z$$ Thus $L(p_1+p_2)$ is a complex vector space of dimension $3$ with basis $\{1,z,\frac 1z\}$.
(You definitely don't need Riemann-Roch, a difficult theorem, to solve this question.)