Dual basis vectors and Basis one-forms I'm studying Tensor Calculus on some MIT's notes (page 16) and I'm stuck at the point where it defines dual basis vectors. I have already studied basis one forms and I can't understand why we need to define another dual space and how these dual basis vectors differs from the basis one-forms.
Thanks in advance.
 A: $\newcommand{\Reals}{\mathbf{R}}\newcommand{\Basis}{\mathbf{e}}$Let $V$ be a real vector space with basis $(\vec{\Basis}_{\mu})_{\mu=1}^{n}$, and assume $V$ is equipped with an inner product $g$ (non-degenerate symmetric bilinear pairing).
The author of the notes defines two isomorphisms from $V$ to the dual space $V^{*}$: On page 8, equation (13) he defines the "linear algebraic" isomorphism (my term) that associates to $\vec{\Basis}_{\mu}$ the basis one-form $\tilde{\Basis}^{\mu}$ defined by
$$
\tilde{\Basis}^{\mu}(\vec{\Basis}_{\nu}) = \delta_{\nu}^{\mu}.
$$
On page 16, equation (42), by contrast, he defines the "geometric" isomorphism (again, my term) that associates to $\vec{\Basis}_{\mu}$ the dual basis vector $\vec{\Basis}^{\mu}$ defined by
$$
\vec{\Basis}^{\mu}(\vec{\Basis}_{\nu}) = g(\vec{\Basis}_{\mu}, \vec{\Basis}_{\nu}).
$$
Both are covectors, but because the metric has spacetime signature (rather than being positive-definite), the two definitions never coincide, even in flat spacetime.
(The tilde and arrow accents are not standard mathematical notation. I've made a good faith effort to match the notation of the notes, but if there's a mismatch, it's probably my fault, not Bertschinger's.)
