I'm reading about the line bundles on projective spaces and there is something that I couldn't make sense of. Maybe I'm missing some definition so please correct me.
Over $\mathbb{P}_k^1$, I know that $\mathscr{O}(-1)=\mathscr{O}(1)^\vee$, and $\Gamma(\mathbb{P}_k^1,\mathscr{O}(1))$ can be identified with the degree 1 polynomials in $x_0,x_1$, but there is no nonzero global section for $\mathscr{O}(-1)$.
But as definition, $$\Gamma(\mathbb{P}_k^1,\mathscr{O}(-1))=\Gamma(\mathbb{P}_k^1,\mathscr{O}(1))^\vee=Hom_{\Gamma(\mathbb{P}_k^1,\mathscr{O}_{\mathbb{P}_k^1})}(\Gamma(\mathbb{P}_k^1,\mathscr{O}(1)),\Gamma(\mathbb{P}_k^1,\mathscr{O}_{\mathbb{P}_k^1})).$$ Here $\Gamma(\mathbb{P}_k^1,\mathscr{O}_{\mathbb{P}_k^1})\simeq k$, and as $k$-module $\Gamma(\mathbb{P}_k^1,\mathscr{O}(1))\simeq k^2$. Then how could $\Gamma(\mathbb{P}_k^1,\mathscr{O}(-1))$ be zero?