I am reading a proof in algebraic geometry where the following equality is used:
$\text{dim} \: L(C,K_C) = \text{dim} \: L(C,K_C + P),$
where $L(C,D)$ is the vector space of meromorphic functions with specified poles on a compact Riemann surface or a non-singular projective algebraic curve, $D$ is a divisor, $K_C$ is a canonical divisor of $C$ and $P$ is a point which belongs to $C$.
In more usual notation, the identity would be
$l(K) = l(K+P).$
Is there some simple way to see that this equality is true (perhaps some complex analytic argument), it seems intuitively obvious to me.