English
The Frobenius norm furnishes a Normed Additive Commutative Group structure on matrices with entries in a normed additively commutative group, via a suitable identification (PiLp norm).
Русский
Для матриц над нормированным абелевой群ой через нормированную норму Фробениуса задается структура нормированного аддитивного коммутативного множества на матрицах.
LaTeX
$$$\text{FrobeniusNormedAddCommGroup} : \mathsf{Matrix}(m,n,\alpha) \text{ is a NormedAddCommGroup whenever } \alpha\text{ is NormedAddCommGroup}$$$
Lean4
/-- Normed group instance (using the Frobenius norm) for matrices over a normed group. Not
declared as an instance because there are several natural choices for defining the norm of a
matrix. -/
@[local instance]
def frobeniusNormedAddCommGroup [NormedAddCommGroup α] : NormedAddCommGroup (Matrix m n α) :=
(by infer_instance : NormedAddCommGroup (PiLp 2 fun i : m => PiLp 2 fun j : n => α))
