Results on direct sums and finitely supported functions.

1. The linear equivalence between finitely supported functions ι →₀ M and the direct sum of copies of M indexed by ι.

noncomputable def finsuppLEquivDirectSum (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] (ι : Type u_1) [DecidableEq ι] : (ι →₀ M) ≃ₗ[R] DirectSum ι fun (x : ι) => M

The finitely supported functions ι →₀ M are in linear equivalence with the direct sum of copies of M indexed by ι.

Equations

• finsuppLEquivDirectSum R M ι = finsuppLequivDFinsupp R

@[simp]

theorem finsuppLEquivDirectSum_single (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] (ι : Type u_1) [DecidableEq ι] (i : ι) (m : M) : (finsuppLEquivDirectSum R M ι) (Finsupp.single i m) = (DirectSum.lof R ι (fun (i : ι) => M) i) m

@[simp]

theorem finsuppLEquivDirectSum_apply (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] (ι : Type u_1) [DecidableEq ι] (m : ι →₀ M) (i : ι) : ((finsuppLEquivDirectSum R M ι) m) i = m i

@[simp]

theorem finsuppLEquivDirectSum_symm_lof (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] (ι : Type u_1) [DecidableEq ι] (i : ι) (m : M) : (finsuppLEquivDirectSum R M ι).symm ((DirectSum.lof R ι (fun (x : ι) => M) i) m) = Finsupp.single i m

theorem lmap_finsuppLEquivDirectSum_eq (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] (ι : Type u_1) [DecidableEq ι] {N : Type u_2} [AddCommMonoid N] [Module R N] (ε : M →ₗ[R] N) (m : ι →₀ M) : (DirectSum.lmap fun (x : ι) => ε) ((finsuppLEquivDirectSum R M ι) m) = (finsuppLEquivDirectSum R N ι) (Finsupp.mapRange ⇑ε ⋯ m)