Tangent vectors and tangent spaces
The set of tangent vectors at p ∈ M is a vector space called the
tangent space, denoted by T
p
M. Indeed, if v, w are tangent
vectors at p and λ ∈ R, then
1 (v + w)(f) = v(f) + w(f ),
2 (λ v)(f ) = λ v(f).
In a local chart, the vectors
∂
∂x
1
p
, . . . ,
∂
∂x
m
p
constitute a basis of T
p
M.