$C^\infty(E)$ is a Frechet VECTOR space. Thus its tangent space at each point equals $C^\infty(E)$ via its affine structure.

#Added:

This is also true if $M$ is not compact. However, for a fiber bundle $Q\to M$ one has to be more careful with the topology (if $M$ is not compact). See 10.10 of

• Peter W. Michor: Manifolds of differentiable mappings. Shiva Mathematics Series 3, Shiva Publ., Orpington, (1980) (pdf)

for an answer. In principle, the tangent space is the space of sections of the vertical bundle of $Q$ restricted to the the image of a section. There is also Sections 42, 43, .. of

• Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997 (pdf)

where a more easily handable notion of differentiability is developped and then used.

These methods also work for Sobolev spaces of sections. See

• MR3135704 Reviewed Inci, H.; Kappeler, T.; Topalov, P. On the regularity of the composition of diffeomorphisms. (English summary) Mem. Amer. Math. Soc. 226 (2013), no. 1062, vi+60 pp.

for the basics of these.

$C^\infty(E)$ is a Frechet VECTOR space. Thus its tangent space at each point equals $C^\infty(E)$ via its affine structure.

Added:

This is also true if $M$ is not compact. However, for a fiber bundle $Q\to M$ one has to be more careful with the topology (if $M$ is not compact). See 10.10 of

• Peter W. Michor: Manifolds of differentiable mappings. Shiva Mathematics Series 3, Shiva Publ., Orpington, (1980) (pdf)

for an answer. In principle, the tangent space is the space of sections of the vertical bundle of $Q$ restricted to the the image of a section. There is also Sections 42, 43, .. of

• Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997 (pdf)

where a more easily handable notion of differentiability is developped and then used.

These methods also work for Sobolev spaces of sections. See

• MR3135704 Reviewed Inci, H.; Kappeler, T.; Topalov, P. On the regularity of the composition of diffeomorphisms. (English summary) Mem. Amer. Math. Soc. 226 (2013), no. 1062, vi+60 pp.

for the basics of these.

$C^\infty(E)$ is a Frechet VECTOR space. Thus its tangent space at each point equals $C^\infty(E)$ via its affine structure.

$C^\infty(E)$ is a Frechet VECTOR space. Thus its tangent space at each point equals $C^\infty(E)$ via its affine structure.

#Added:

This is also true if $M$ is not compact. However, for a fiber bundle $Q\to M$ one has to be more careful with the topology (if $M$ is not compact). See 10.10 of

• Peter W. Michor: Manifolds of differentiable mappings. Shiva Mathematics Series 3, Shiva Publ., Orpington, (1980) (pdf)

for an answer. In principle, the tangent space is the space of sections of the vertical bundle of $Q$ restricted to the the image of a section. There is also Sections 42, 43, .. of

• Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997 (pdf)

where a more easily handable notion of differentiability is developped and then used.

These methods also work for Sobolev spaces of sections. See

• MR3135704 Reviewed Inci, H.; Kappeler, T.; Topalov, P. On the regularity of the composition of diffeomorphisms. (English summary) Mem. Amer. Math. Soc. 226 (2013), no. 1062, vi+60 pp.

for the basics of these.