Rational lattice

Positive definite lattice

A rational lattice is said to be positive definite iff for any nonzero . #m/def/geo Equivalently, the quadratic space is (“Euclidean space”).1

Properties

  1. There exist finitely many lattice points of a given norm, i.e. for any .
  2. Assume is integral and . Then and
Proof of 1–2

Since

where is compact and is discrete, it follows is finite, proving ^P2.

^P2 follows from the Cauchy-Schwarz inequality.

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 1988. Vertex operator algebras and the Monster, §6.1, pp. 122–124 ↩