Positive definite lattice

A rational lattice $𝐿$ is said to be positive definite iff $⟨ 𝛼, 𝛼 ⟩ > 0$ for any nonzero $𝛼 \in 𝐿$ . geo Equivalently, the quadratic space $𝐿 ℝ$ is $ℝ 𝑛, 0$ (“Euclidean space”).¹

Properties

There exist finitely many lattice points of a given norm, i.e. $| 𝐿 𝑚 | < \infty$ for any $𝑚 \in ℚ$ .
Assume $𝐿$ is ^integral and $𝛼, 𝛽 \in 𝐿 2$ . Then $⟨ 𝛼, 𝛽 ⟩ \in {0, \pm 1, \pm 2}$ and

⟨ 𝛼, 𝛽 ⟩ = - 2 ⟺ 𝛼 + 𝛽 = 0 ⟨ 𝛼, 𝛽 ⟩ = - 1 ⟺ 𝛼 + 𝛽 \in 𝐿 2

Proof of 1–2

Since
$𝐿 𝑚 = B 𝑚 (⃗ 𝟎) \cap 𝐿$
where $B 𝑚 (⃗ 𝟎)$ is compact and $𝐿$ is discrete, it follows $𝐿 𝑚$ is finite, proving ^P2.

^P2 follows from the Cauchy-Schwarz inequality.