English
From a DualBases h, we build a basis h.basis of M by pairing coeffs and lc as left and right inverses; i.e., h.basis := Basis.ofRepr { toFun := coeffs h, invFun := lc e, left_inv := lc_coeffs h, right_inv := coeffs_lc h, ... }
Русский
Из DualBases h строится база h.basis модуля M через отображения coeffs и lc как пары левых и правых обратных; то есть h.basis = Basis.ofRepr { ... }
LaTeX
$$$\\text{basis} := \\text{Basis.ofRepr}\\{ toFun := coeffs\\, h, \\; invFun := lc\\, e, \\; \\text{left_inv} := lc\\_coeffs\\, h, \\; \\text{right_inv} := coeffs\\_lc\\, h \\}$$$
Lean4
@[simp]
theorem coeffs_lc (l : ι →₀ R) : h.coeffs (DualBases.lc e l) = l :=
by
ext i
rw [h.coeffs_apply, h.dual_lc]
