Linear form/Analysis/Example

Many important examples of linear forms on some vector spaces of infinite dimension arise in analysis. For a real interval ${\displaystyle {}[a,b]}$, the set of functions ${\displaystyle {}\operatorname {Map} \,{\left([a,b],\mathbb {R} \right)}}$, or the set of continuous functions ${\displaystyle {}C([a,b],\mathbb {R} )}$, or the set of continuously differentiable functions ${\displaystyle {}C^{1}([a,b],\mathbb {R} )}$ form real vector spaces. For a point ${\displaystyle {}P\in [a,b]}$, the evaluation ${\displaystyle {}f\mapsto f(P)}$ is a linear form (because addition and scalar multiplication is defined pointwisely on these spaces). Also, the evaluation of the derivative at ${\displaystyle {}P}$,

${\displaystyle C^{1}([a,b],\mathbb {R} )\longrightarrow \mathbb {R} ,f\longmapsto f'(P),}$

is a linear form. For ${\displaystyle {}C([a,b],\mathbb {R} )}$, the integral, that is, the mapping

${\displaystyle C([a,b],\mathbb {R} )\longrightarrow \mathbb {R} ,f\longmapsto \int _{a}^{b}f(t)dt,}$

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