Orthogonality defect

Exercise 1.1 (Hadamard inequality)   Show that if $\{{\mathbf{b}}_1,\cdots {\mathbf{b}}_n\}$ is a basis of $\mathbb{R}^n$, then

$$\vert$$det$$({\mathbf{b}}_1\cdots {\mathbf{b}}_n)\vert\leq\prod_{i=1}^n \Vert{\mathbf{b}}_i\Vert$$

Exercise 1.2   Show that in the previous exercise, there is equality if and only if the basis is orthogonal.

Definition 1.3   The orthogonality defect of a basis $\{{\mathbf{b}}_1,\cdots {\mathbf{b}}_n\}$ is defined as the quantity:

$$\frac{\prod_{i=1}^n \Vert{\mathbf{b}}_i\Vert}{\vert\text{det}({\mathbf{b}}_1\cdots {\mathbf{b}}_n)\vert}$$

The following theorem is left as a challenge to the reader.

Theorem 1.4   Every lattice has a basis with orthogonality defect less than $n^n$.