[Merged by Bors] - feat: OrderIso between finite-codimensional subspaces and finite-dimensional subspaces in the dual #9124
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…nsional subspaces in the dual
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| theorem quotDualCoannihilatorToDual_nondegenerate (W : Submodule R (Dual R M)) : | ||
| W.quotDualCoannihilatorToDual.Nondegenerate := by |
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I'm amazed that this pairing is always nondegenerate for any submodule of any module over any commutative ring. One half of both the nondegeneracy of Submodule.dualPairing and of Submodule.dualCopairing require that the module is over a field. M*/W → coann(W) may not even be injective even if M is a vector space.
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…nsional subspaces in the dual (#9124) + Introduce the nondegenerate pairing (`(flip_)quotDualCoannihilatorToDual_injective`) between `M ⧸ W.dualCoannihilator` and `W` . If `M` is a vector space and `W` is a finite-dimensional subspace of its dual, this is a perfect pairing (`quotDualCoannihilatorToDual_bijective`), and `W` is equal to the annihilator of its coannihilator. + Use this pairing to show that `dualAnnihilator` and `dualCoannihilator` give an antitone order isomorphism `orderIsoFiniteCodimDim` between finite-codimensional subspaces in a vector space and finite-dimensional subspaces in its dual. This result can be e.g. found in Bourbaki's Algebra. For a finite-dimensional vector space, this gives an OrderIso between all subspaces and all subspaces of the dual. + Add some lemmas about the image and preimage of annihilators and coannihilators under `Dual.eval`. + Expand the docstring of `basis_finite_of_finite_spans` with comments on generalizations. - [x] depends on: #8820 Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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| -- or over a ring satisfying the strong rank condition | ||
| -- (which covers all commutative rings; see `LinearIndependent.finite_of_le_span_finite`). | ||
| -- this is not true in general. |
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Introduce the nondegenerate pairing (
(flip_)quotDualCoannihilatorToDual_injective) betweenM ⧸ W.dualCoannihilatorandW. IfMis a vector space andWis a finite-dimensional subspace of its dual, this is a perfect pairing (quotDualCoannihilatorToDual_bijective), andWis equal to the annihilator of its coannihilator.Use this pairing to show that
dualAnnihilatoranddualCoannihilatorgive an antitone order isomorphismorderIsoFiniteCodimDimbetween finite-codimensional subspaces in a vector space and finite-dimensional subspaces in its dual. This result can be e.g. found in Bourbaki's Algebra. For a finite-dimensional vector space, this gives an OrderIso between all subspaces and all subspaces of the dual.Add some lemmas about the image and preimage of annihilators and coannihilators under
Dual.eval.Expand the docstring of
basis_finite_of_finite_spanswith comments on generalizations.