[Merged by Bors] - feat(LinearAlgebra/PiTensorProduct): add basis for PiTensorProduct and map for the tensor product of dual space#32613
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Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
…community/mathlib4 into SM.PiTensorProduct.DirectSum
…community/mathlib4 into SM.PiTensorProduct.DirectSum
Co-authored-by: Eric Wieser <efw@google.com>
Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>
PR summary ef704c53bfImport changes for modified filesNo significant changes to the import graph Import changes for all files
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Excellent work, thanks!
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…and map for the tensor product of dual space (#32613) Constructs a basis for `PiTensorProduct` given bases for the component spaces and defines maps between the tensor product of dual spaces and the dual of a tensor product. Main definitions: * `Basis.piTensorProduct`: Given bases `b : Π i, Basis (κ i) R (M i)` for each component space `M i`, constructs a basis for `⨂[R] i, M i` indexed by `Π i, κ i` defined by sending `p : Π i, κ i` to `⨂ₜ[R] i, (b i) (p i)`. * `PiTensorProduct.dualDistrib`: The canonical linear map from `⨂[R] i, Dual R (M i)` to `Dual R (⨂[R] i, M i)`, sending `⨂ₜ[R] i, f i` to the composition of `PiTensorProduct.map f` with the linear equivalence `⨂[R] i, R →ₗ R` given by multiplication. * `PiTensorProduct.dualDistribEquiv`: A linear equivalence between `⨂[R] i, Dual R (M i)` and `Dual R (⨂[R] i, M i)` when all `M i` are finite free modules. If `f : (i : ι) → Dual R (M i)`, then this equivalence sends `⨂ₜ[R] i, f i` to the composition of `PiTensorProduct.map f` with the natural isomorphism `⨂[R] i, R ≃ R` given by multiplication. - [x] depends on: #32456
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PiTensorProduct and map for the tensor product of dual spacePiTensorProduct and map for the tensor product of dual space
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Constructs a basis for
PiTensorProductgiven bases for the component spaces and defines maps between the tensor product of dual spaces and the dual of a tensor product.Main definitions:
Basis.piTensorProduct: Given basesb : Π i, Basis (κ i) R (M i)for each component spaceM i, constructs a basis for⨂[R] i, M iindexed byΠ i, κ idefined by sendingp : Π i, κ ito⨂ₜ[R] i, (b i) (p i).PiTensorProduct.dualDistrib: The canonical linear map from⨂[R] i, Dual R (M i)toDual R (⨂[R] i, M i), sending⨂ₜ[R] i, f ito the composition ofPiTensorProduct.map fwith the linear equivalence⨂[R] i, R →ₗ Rgiven by multiplication.PiTensorProduct.dualDistribEquiv: A linear equivalence between⨂[R] i, Dual R (M i)and
Dual R (⨂[R] i, M i)when allM iare finite free modules. Iff : (i : ι) → Dual R (M i), then this equivalence sends⨂ₜ[R] i, f ito the composition ofPiTensorProduct.map fwith the natural isomorphism⨂[R] i, R ≃ Rgiven by multiplication.PiTensorProducts overFinsupp,DFinsupp, andDirectSum#32456