Theorems · Definition · linear algebra

LinearMap.lsum

(R : Type u) →
  {M : Type v} →
    {ι : Type x} →
      [inst : Semiring R] →
        (φ : ι → Type i) →
          [inst_1 : (i : ι) → AddCommMonoid (φ i)] →
            [inst_2 : (i : ι) → Module R (φ i)] →
              [DecidableEq ι] →
                (S : Type u_1) →
                  [inst_4 : AddCommMonoid M] →
                    [inst_5 : Module R M] →
                      [Fintype ι] →
                        [inst_7 : Semiring S] →
                          [inst_8 : Module S M] →
                            [inst_9 : SMulCommClass R S M] → ((i : ι) → φ i →ₗ[R] M) ≃ₗ[S] ((i : ι) → φ i) →ₗ[R] M

The linear equivalence between linear functions on a finite product of modules and families of functions on these modules. See note [bundled maps over different rings].

Defined in

Mathlib.LinearAlgebra.Pi

Cited by

10 results in Mathlib

Foundations

Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModuleDecidableEqAddCommMonoidModuleFintypeSemiringModuleSMulCommClass