Let $n \in \mathbb{Z}_{\ge 0}$, let $x_0,x_1,\dots,x_n \in \mathbb{R}$ be pairwise distinct, and let $A \subset \mathbb{R}$ satisfy $\{x_0,x_1,\dots,x_n\} \subset A$. Let $\mathcal{P}_n(\mathbb{R})$ denote the real [vector space](/page/Vector%20Space) of polynomial functions $p:\mathbb{R}\to\mathbb{R}$ with degree at most $n$. Define the interpolation operator

\begin{align*} I_{x_0,\dots,x_n}: \{f:A\to\mathbb{R}\} \to \mathcal{P}_n(\mathbb{R}) \end{align*}

by requiring that $I_{x_0,\dots,x_n}f$ is the unique polynomial function in $\mathcal{P}_n(\mathbb{R})$ satisfying

\begin{align*} (I_{x_0,\dots,x_n}f)(x_i)=f(x_i) \end{align*}

for every $i \in \{0,1,\dots,n\}$. Then for all functions $f,g:A\to\mathbb{R}$ and all scalars $\alpha,\beta\in\mathbb{R}$,

\begin{align*} I_{x_0,\dots,x_n}(\alpha f+\beta g)=\alpha I_{x_0,\dots,x_n}f+\beta I_{x_0,\dots,x_n}g \end{align*}

as elements of $\mathcal{P}_n(\mathbb{R})$.