Lattices

In Chapter 2, we briefly introduced lattices in the Euclidean space \(\mathbb {R}^{n}\) . We had defined basis and determinant of a lattice. Here we derive some more results on the choice of basis of a lattice. The notion of sublattice and its index is introduced. As applications, we prove the results on the representation of a positive integer as a sum of two squares and a sum of four squares. We also discuss convergence and boundedness of a sequence of lattices and give a proof of Mahler’s compactness theorem.