Assuming a three-dimensional Bravais lattice and labelling each lattice vector (a vector indicating a lattice point) by the subscript  as 3-tuple of integers,
: where 
where  is the set of integers and  is a primitive translation vector or shortly primitive vector. Taking a function  where  is a position vector from the origin  to any position, if  follows the periodicity of this lattice, e.g. the function describing the electronic density in an atomic crystal, it is useful to write  as a multi-dimensional Fourier series
where now the subscript , so this is a triple sum.
As  follows the periodicity of the lattice, translating  by any lattice vector  we get the same value, hence
Expressing the above instead in terms of their Fourier series we have
Because equality of two Fourier series implies equality of their coefficients, , which only holds when
: where 
Mathematically, the reciprocal lattice is the set of all vectors , that are wavevectors of plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice as the set of all direct lattice point position vectors , and  satisfy this equality for all . Each plane wave in the Fourier series has the same phase (actually can be differed by a multiple of ) at all the lattice point .
As shown in the section multi-dimensional Fourier series,  can be chosen in the form of  where . With this form, the reciprocal lattice as the set of all wavevectors  for the Fourier series of a spatial function which periodicity follows , is itself a Bravais lattice as it is formed by integer combinations of its own primitive translation vectors , and the reciprocal of the reciprocal lattice is the original lattice, which reveals the Pontryagin duality of their respective vector spaces. (There may be other form of . Any valid form of  results in the same reciprocal lattice.)