Editing Linear form (section)

=== Evaluation ===
Let <math>P_n</math> denote the vector space of real-valued polynomial functions of degree <math>\leq n</math> defined on an interval <math>[a, b].</math> If <math>c \in [a, b],</math> then let <math>\operatorname{ev}_c : P_n \to \R</math> be the '''evaluation functional'''
<math display=block>\operatorname{ev}_c f = f(c).</math>
The mapping <math>f \mapsto f(c)</math> is linear since
<math display=block>\begin{align}
     (f + g)(c) &= f(c) + g(c) \\
  (\alpha f)(c) &= \alpha f(c).
\end{align}</math>

If <math>x_0, \ldots, x_n</math> are <math>n + 1</math> distinct points in <math>[a, b],</math> then the evaluation functionals <math>\operatorname{ev}_{x_i},</math> <math>i = 0, \ldots, n</math> form a [[Basis of a vector space|basis]] of the dual space of <math>P_n</math> ({{harvtxt|Lax|1996}} proves this last fact using [[Lagrange interpolation]]).