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.