A dual basis of a vector basis 
, ..., 
of a finite-dimensional vector space 
over a field 
is the vector basis 
, ..., 
of the dual vector space 
characterized by
 |
(1)
|
where 
is the Kronecker
delta. These linear functionals extract
the coordinates of a vector,
so 
(Axler 2026,
pp. 106-107). No metric tensor is needed to
define a dual basis.
The elements 
are covectors, rather than
vectors in 
. In tensor terminology, the latter are contravariant
tensors of tensor rank 1, whereas covectors
are covariant tensors of tensor
rank 1. A symmetric bilinear form 
on a real
vector space that is nondegenerate
identifies covectors with representing vectors 
satisfying
 |
(2)
|
For the dot product in Euclidean space, an orthonormal basis satisfies
 |
(3)
|
so the representing vectors coincide with the original vector basis. This is an identification using the chosen inner product, not equality between vectors and linear functionals as different types of objects.
For a Hermitian inner product linear in its first argument, the corresponding representation is 
. The map from representing
vectors to linear functionals
is then antilinear (Axler 2026, p. 216).
