Let $C$ be a smooth irreducible projective algebraic curve over an algebraically closed field.

Let $D$ be an effective divisor on $C$. Prove that the vector space $L(D)$ of rational functions with poles bounded by $D$ is finite dimensional.

Let $D$ and $E$ be linearly equivalent divisors on $C$. Exhibit an isomorphism between the vector spaces $L(D)$ and $L(E)$.

What is a canonical divisor on $C$ ? State the Riemann-Roch theorem and use it to calculate the degree of a canonical divisor in terms of the genus of $C$.

Prove that the canonical divisor on a smooth cubic plane curve is linearly equivalent to the zero divisor.