Suppose we have a coordinate system . This defines a coordinate basis , where for each the basis vector has components
in these coordinates. We also have the coordinate dual basis where each dual vector or “1-form” has components
Now while these bases are dual in the sense of bases:
(by definition of dual basis), the individual vectors are not dual to the individual 1-forms in the sense of individual vectors. That is, for any given , we have and are not dual in general.
Instead, recall indices are raised and lowered using the metric components (in a coordinate basis, that is). Possibly the result could be seen by inspection, but for clarity let’s write for some chosen . This vector has components , hence the corresponding 1-form has components . By the meaning of components this says the 1-form is . This is not , in general! In “musical isomorphism” notation, the result is:
To show the result another way, recall the metric defines the dual to our vector to be . To examine this 1-form, feed it a vector (specifically, basis vectors ) and see how it acts on it:
which says , as before.
In closing, another reason we cannot have in general is that the coordinate basis vector is not defined in terms of alone, but also all the other coordinates chosen. More on that next.
(Schutz (2009, §3.3, §3.5) makes a superb background to this discussion, and while the cited sections are for special relativity, in this case you can simply replace the Minkowski metric with an arbitrary curved metric .)