$L(D)$ is Vector Space

Question

Given a divisor $D$ on a curve $X$, define $L(D)=\{0\}\cup \{f \in k(X),f\ne 0 \, | (f)+D \ge 0\}$.

where $(f)=\sum \nu_P(f)P$ and $ \upsilon_{P}(f)= |zeros| − |poles| $ of $f$ at $P$. I want to show that $L(D)$ is a vector space, any help appreciated.

Answer 1

$v_P(f+g)\geq \operatorname {min}(v_P(f),v_P(g))\quad \quad v_P(\lambda f)=v_P(f) _\;(\lambda \neq0)$