From Online Dictionary of Crystallography
Base duale (Fr). Base duale (It)
The dual basis is a basis associated to the basis of a vector space. In three-dimensional space, it is isomorphous to the basis of the reciprocal lattice. It is mathematically defined as follows:
Given a basis of n vectors ei spanning the direct space En, and a vector x = x i ei, let us consider the n quantities defined by the scalar products of x with the basis vectors, ei:
xi = x . ei = x j ej . ei = x j gji,
where the gji 's are the doubly covariant components of the metric tensor.
By solving these equations in terms of x j, one gets:
x j = xi gij
where the matrix of the gij 's is inverse of that of the gij 's (gikgjk = δij). The development of vector x with respect to basis vectors ei can now also be written:
x = x i ei = xi gij ej
The set of n vectors ei = gij ej that span the space En forms a basis since vector x can be written:
x = xi ei
This basis is the dual basis and the n quantities xi defined above are the coordinates of x with respect to the dual basis. In a similar way one can express the direct basis vectors in terms of the dual basis vectors:
ei = gij ej
The scalar products of the basis vectors of the dual and direct bases are:
gij = ei . ej = gik ek . ej = gikgjk = δij.
One has therefore, since the matrices gik and gij are inverse:
gij = ei . ej = δij.
These relations show that the dual basis vectors satisfy the definition conditions of the reciprocal vectors. In a three-dimensional space the dual basis and the basis of reciprocal space are identical.
Change of basis
In a change of basis where the direct basis vectors and coordinates transform like:
e'j = Aji ei; x'j = Bi j xi,
where Aji and Bi j are transformation matrices, transpose of one another, the dual basis vectors ei and the coordinates xi transform according to:
e'j = Bi j ei; x'j = Ajixi.
The coordinates of a vector in reciprocal space are therefore covariant and the dual basis vectors (or reciprocal vectors) contravariant.
The Reciprocal Lattice (Teaching Pamphlet of the International Union of Crystallography)
Section 1.1.2 of International Tables of Crystallography, Volume D