Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $n\in\mathbb N$, let $X \subseteq \mathbb A_k^n$ be an affine variety with the subspace Zariski topology, and let $R:=k[x_1,\dots,x_n]$. Let $I(X) \trianglelefteq R$ be the vanishing ideal of $X$, and let $\mathcal O(X)$ denote the $k$-algebra of regular functions $f:X\to k$, meaning functions such that for every $p\in X$ there are an open neighbourhood $U\subseteq X$ of $p$ and polynomials $A,B\in R$ with $B(q)\ne 0$ and $f(q)=A(q)/B(q)$ for every $q\in U$. Define

\begin{align*} \Phi: R/I(X) &\longrightarrow \mathcal O(X) \end{align*}

by

\begin{align*} \overline{F} &\longmapsto \left(p \mapsto F(p)\right). \end{align*}

Then $\Phi$ is an isomorphism of $k$-algebras.