Many important examples of linear forms on some vector spaces of infinite dimension arise in analysis. For a real interval ![{\displaystyle {}[a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e5a6e660b5cdc1a9c76f6e6b0d486328c3f6937)
, the set of functions ![{\displaystyle {}\operatorname {Map} \,{\left([a,b],\mathbb {R} \right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1927ffc4862705b88b61a0f193731cd35438ccb6)
, or the set of continuous functions ![{\displaystyle {}C([a,b],\mathbb {R} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25d63a9ada20e090e56ad8f428af4742361c746d)
, or the set of continuously differentiable functions ![{\displaystyle {}C^{1}([a,b],\mathbb {R} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9769bb0542a28577c2ad0ae210db48ceda9f90c)
form real vector spaces. For a point
![{\displaystyle {}P\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8450ee0b2622432387847a2450ee13d62fb3107d)
,
the evaluation 
is a linear form
(because addition and scalar multiplication is defined pointwisely on these spaces).
Also, the evaluation of the derivative at 
,
-
![{\displaystyle C^{1}([a,b],\mathbb {R} )\longrightarrow \mathbb {R} ,f\longmapsto f'(P),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/568491cc0424846b5fec60fa018aa1c62d1f5b56)
is a linear form. For , the integral, that is, the mapping
-
![{\displaystyle C([a,b],\mathbb {R} )\longrightarrow \mathbb {R} ,f\longmapsto \int _{a}^{b}f(t)dt,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5360e55acd7062980c4462189a269a5fc3cf82ee)
is a linear form. This rests on the linearity of the integral.
