I want to show smooth manifold has infinite dimensional space of smooth vector field. I know the space of smooth function of a smooth manifold is infinite dimensional.
Then can I just choose a local chart and consider V(p)=f(p)$\frac{\partial} {\partial x}$, and claim that since space of smooth function is infinite dimensional, then so is the space of smooth vector field. I mean if f_1 and f_2 are linear independent, $f_1\frac{\partial}{\partial x}$ and $f_2\frac{\partial}{\partial x}$ should be linear independent.