English
A bilinear form B is nondegenerate if the only vector m with B(m,n)=0 for all n is m=0.
Русский
Билинейная форма B неdegenerate, если единственный вектор m, для которого B(m,n)=0 для всех n, равен нулю.
LaTeX
$$def Nondegenerate (B : BilinForm R M) : Prop := ∀ m : M, (∀ n : M, B m n = 0) → m = 0$$
Lean4
/-- A nondegenerate bilinear form is a bilinear form such that the only element that is orthogonal
to every other element is `0`; i.e., for all nonzero `m` in `M`, there exists `n` in `M` with
`B m n ≠ 0`.
Note that for general (neither symmetric nor antisymmetric) bilinear forms this definition has a
chirality; in addition to this "left" nondegeneracy condition one could define a "right"
nondegeneracy condition that in the situation described, `B n m ≠ 0`. This variant definition is
not currently provided in mathlib. In finite dimension either definition implies the other. -/
def Nondegenerate (B : BilinForm R M) : Prop :=
∀ m : M, (∀ n : M, B m n = 0) → m = 0
