Theorems · Definition · linear algebra

Finsupp.lsum

{α : Type u_1} →
  {M : Type u_2} →
    {N : Type u_3} →
      {R : Type u_5} →
        {R₂ : Type u_6} →
          (S : Type u_8) →
            [inst : Semiring R] →
              [inst_1 : Semiring R₂] →
                [inst_2 : Semiring S] →
                  [inst_3 : AddCommMonoid M] →
                    [inst_4 : Module R M] →
                      [inst_5 : AddCommMonoid N] →
                        [inst_6 : Module R₂ N] →
                          {σ : R →+* R₂} →
                            [inst_7 : Module S N] →
                              [inst_8 : SMulCommClass R₂ S N] → (α → M →ₛₗ[σ] N) ≃ₗ[S] (α →₀ M) →ₛₗ[σ] N

Lift a family of linear maps M →ₗ[R] N indexed by x : α to a linear map from α →₀ M to N using Finsupp.sum. This is an upgraded version of Finsupp.liftAddHom. See note [bundled maps over different rings] for why separate R and S semirings are used.

Defined in

Mathlib.LinearAlgebra.Finsupp.LSum

Cited by

40 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringSemiringAddCommMonoidModuleAddCommMonoidModuleModuleSMulCommClass