Linear form/Analysis/Example

Many important examples of linear forms on some vector spaces of infinite dimension arise in analysis. For a real interval , the set of functions , or the set of continuous functions , or the set of continuously differentiable functions form real vector spaces. For a point , the evaluation is a linear form (because addition and scalar multiplication is defined pointwisely on these spaces). Also, the evaluation of the derivative at ,

is a linear form. For , the integral, that is, the mapping

is a linear form. This rests on the linearity of the integral.