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.

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_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) : (LinearEquiv.symm (finsuppLEquivDirectSum R M ι)) ((DirectSum.lof R ι (fun (x : ι) => M) i) m) = Finsupp.single i m