Divisors on the complex plane

Question

I am trying to understand the notion of $L(D) = \{f \text{ is meromorphic}| D + (f) \geq 0\}$ on $C\mathbb{P}^1$. I have come across an exercise where it is asked to prove that $dim(L(D))=\max{(0,1+deg(D))}$ for any divisors $D$ on $C\mathbb{P}^1$. Let $D = \sum_i n_i P_i$, if $deg(D)<0$, it means that $\sum_i n_i < 0$. I know that a meromorphic function $f$ on $C\mathbb{P}^1$ admits a decomposition as a rational function which means that $(f) = -\sum_i^e n_i \cdot \infty + \sum_i^e n_i x_i $. By the linearity of the degree, I get that $deg(D + (f)) = deg(D) + deg((f)) = deg(D) < 0$ and so there is no $f$ meromorphic in that case so $dim(L(D))$ is indeed equal to $0$. In the case where $deg(D) \geq 0$, I have trouble translating this into $L(D)$. For example $deg(D) =0$, then I can represent $D$ as a meromorphic function, i.e. $D = (g)$ then $D+(f) = (fg) \geq 0$ is always true for any meromorphic function as the product of two rational function yields another rational function. This means that the $dim(L(D))$ is not equal to 1. I feel like there is something I misunderstood but I cannot pinpoint what it is. Thank you for your help!