DiffGeo

where [math] \left( \frac{ \partial }{ \partial x^i} \right) : C^{\infty} (M) \longrightarrow \mathbb{R}[/math]

>>9157740
because dx^j/dx^i = 1 when i = j and 0 otherwise.

>>9157744
so what? i mean [math]\left( \frac{ \partial}{ \partial x^i} \right)_p[/math] is just an operator. How in the world you can write every [math]X \in T_p M[/math] as [math]X=X_i \left( \frac{ \partial}{ \partial x^i} \right)_p [/math]??

>>9157740
The answer is to completely disregard Differential Geometry.

>>9157757
why is that?

>>9157751
TpM is a space of operators, a vector in it is actually (kind of) a functional you apply to the base point. thats what the inner product x^ix_i is, its actualy the operator x_i you apply to the point x^i and get the real value x_ix^i (change accordingly depending on what dimension the different vectors are).

>>9157860
oh.. so its got something to do with [math]T^{\ast }_p M[/math] ? since you said functional..