Global regular functions on projective space ($\Gamma ({\mathbb{P}}_k^n,\mathcal{O})=k$)

The only regular functions defined on all of a nonempty projective space over a field $k$ are the constants:For ${\mathbb{P}}^1$, this also follows by gluing $k[t]$ and $k[t^{-1}]$ inside $k[t,t^{-1}]$: their intersection is $k$.