I'm having trouble understanding the following sentence from Schutz's

[schwarzschild]

I'm having trouble understanding the following sentence from Schutz's A First Course in General Relativity, so I was hoping someone could explain/expound on it. "
$$p_a \equiv \widetilde{p}( \vec{e_{\alpha} ) } )$$ Any component with a single lower index is, by convention, the component of a one-form; an upper index denotes the component of a vector."

 

[haushofer]

If I have a vector X, I can write it in component form as

$$X = X^{\mu}e_{(\mu)}$$

I put round brackets around the mu index of the basis vector e to indicate that for every mu we have a whole vector, not just one component! Now, I can act with this vector X on the dual basis:

$$X(e^{(\nu)}) = X^{\mu} e_{(\mu)} (e^{(\nu)}) = X^{\mu}\delta_{\mu}^{\nu} = X^{\nu}$$
In words: acting with the vector X on the dual basis means acting with the basis on the dual basis, times the components of X. By definition this gives me a delta function times the components of X, which equals the components of X. This holds for every tensor T in general; you could try it for, say, a rank (2,1) tensor T.

Components with lower indices transform different from components with upper indices; the latter we sometimes call "contravariant", the former "covariant" ( co goes below ).