Well, I signed in because I saw a clear case of vandalism in one page that, apparently, had been around for several months. As a regular user of WP, I felt the need to start contributing back, whence, signing in and removing that edit.

Any way, I think now I might stay around and see if I can really contribute with anything useful.

This is a coordinate-free, algebraic characterization of a (pseudo-) metric tensor^[1] .

Let ${\displaystyle \mathbb {E} }$ denote a finite-dimensional linear space over a field ${\displaystyle \mathbf {F} }$ (e.g., ${\displaystyle \mathbb {R} ,\,\mathbb {C} }$), with vectors ${\displaystyle \mathbf {v} \,=\,v^{\alpha }\mathbf {e} _{\alpha }}$, where summation over repeated indices (Einstein convention) is assumed and the set ${\displaystyle \{\mathbf {e} _{\alpha }\}}$ is a basis of ${\displaystyle \mathbb {E} }$. The dual of the space, denoted as ${\displaystyle \mathbb {E^{*}} }$, is the vectorial space of Linear functionals or forms (see also one-form), denoted as ${\displaystyle \mathbf {f} \,=\,f_{\alpha }\mathbf {e} ^{\alpha }}$, that map ${\displaystyle \mathbb {E} }$ into ${\displaystyle \mathbf {F} }$, i.e.,

${\displaystyle \forall \,\mathbf {f} \,\in \,\mathbb {E^{*}} }$

${\displaystyle \mathbf {f} \,:\,\mathbb {E} \longrightarrow \mathbf {F} }$

${\displaystyle \mathbf {v} \mapsto \langle \,\mathbf {f} \,,\,\mathbf {v} \rangle _{\mathbb {E} }\,\equiv \,f_{\alpha }\,v^{\alpha }\,\in \,\mathbf {F} }$

where ${\displaystyle \langle \;,\;\rangle _{\mathbb {E} }}$ will be called the duality product in ${\displaystyle \mathbb {E} }$.
${\displaystyle \{\mathbf {e} ^{\alpha }\}}$ is the basis of ${\displaystyle \mathbb {E^{*}} }$ called the dual basis of ${\displaystyle \{\mathbf {e} _{\alpha }\}}$ if it satisfies that

${\displaystyle \langle \mathbf {e} ^{\alpha }\,,\,\mathbf {e} _{\beta }\rangle \,=\,\delta _{\beta }^{\alpha }}$.

Polnasam (talk) 21:11, 24 June 2011 (UTC)