Riemann-Roch theorem for curves

Let $C$ be a projective nonsingular curve over an algebraically closed field. If $D$ is a divisor on $C$, then

$$\mathrm{\ell }(D)-\mathrm{\ell }(K-D)=\mathrm{deg}(D)+1-g$$

where $g$ is the genus of the curve, and $K$ is the canonical divisor ($\mathrm{\ell }(K)=g$). Here $\mathrm{\ell }(D)$ denotes the dimension of the http://planetmath.org/node/SpaceOfFunctionsAssociatedToADivisorspace of functions associated to a divisor.