Let $\overline{L}$ be a metrized line bundle and let $s$ be a regular meromorphic section of $L$. Its divisor $\operatorname{div}(s)$ is a Cartier divisor in $\mathrm{X}$. The function $\log\left\|{s}\right\|^{-1}$ is defined on the open set $\mathrm{X}\setminus\left|{\operatorname{div}(s)}\right|$ ; by analogy to the complex case, we call it a Green function for the divisor $\operatorname{div}(s)$. When the metric on $\overline{L}$ is smooth, the Green function is said smooth. The same remark applies for the other qualificatives semi-positive, or admissible, that wil be introduced later.

Let us take for $L$ the trivial line bundle, with its canonical trivialization $s=1$, and let us endow it with a smooth metric. By definition, we call $\log\left\|{s}\right\|^{-1}$ a smooth function. More generally, we define the space $\mathscr{C}^{\infty}(\mathrm{X})$ of (real valued) smooth functions to be the real vector space spanned by these elementary smooth functions. Observe that this definition reverses what happens in complex geometry where smooth metrics on the trivial line bundle are defined from the knowledge of smooth functions.