Euclidean norm on integer lattice

Question

Does the Euclidean $ L^2 $ norm (and distance) make any sense on an integer lattice in $ \mathbb{R}^n $? And what is the preferable way of calculating a type of norm in such spaces?

Answer 1

Not sure what you want. There are several useful ways to deal with an integral lattice.

One method is to take a subset of points in ordinary Euclidean space, such as the picture of a triangular sort of lattice in the plane from the Wikipedia article. That is the most popular method. The norm is the squared distance from the origin. The lattice is called integral if all norms are integers.

The older method is to insist on the standard integer points, and assign a positive quadratic form like $x^2 + xy + 2y^2.$ Take all coefficients integers. You can tell a form is positive by taking the Hessian matrix of second partial derivatives, which is symmetric. The form is positive when the Hessian matrix is positive. I prefer this method because I like polynomials and matrices.