\begin{definition}[Definition:Lagrange Basis Polynomial]
Let $x_0, \ldots, x_n \in \R$ be real numbers.
The '''Lagrange basis polynomials''' associated to the $x_i$ are the polynomials:
:$\ds \map {L_j} X := \prod_{\substack {0 \mathop \le i \mathop \le n \\ i \mathop \ne j} } \frac {X - x_i} {x_j - x_i} \in \R \sqbrk X$
{{explain|$\R \sqbrk X$}}
{{mistake|Not sure if it's a mistake or a different way of defining it, but {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|entry = Lagrange interpolation formula}} has this wrapped up in another product symbol}}
{{NamedforDef|Joseph Louis Lagrange|cat = Lagrange}}
\end{definition}