 Solution

For a divisor $D={\sum }_p{n}_{p}p$ on the smooth projective curve,This is a $k$-vector space of rational functions with the prescribed pole bounds. A nonzero rational function on a projective curve has divisor of degree zero. Thus if $\mathrm{deg}D<0$ there can be no nonzero element of $L(D)$.

For any point $p$ with coefficient $n_p$ in $D$, use a local parameter $t$ at $p$. The linear map sending $f\in L(D)$ to the residue-field value of $t^{n_p}f$ has kernel $L(D-p)$ and target $k$, since $k$ is algebraically closed. Therefore $\dim L(D)/L(D-p)\le 1$. If $d=\mathrm{deg}D\ge 0$, successively subtract a point $d+1$ times. The final divisor has degree $-1$ and no nonzero sections, givingEquivalently the universally valid bound is $\dim L(D)\le \max (0,\mathrm{deg}D+1)$. The PDF states the unqualified bound $\dim L(D)\le \mathrm{deg}D+1$; literally this is false when $\mathrm{deg}D\le -2$, because its right side is negative. For example $D=-2p$ has $L(D)=0$.