Theorems · Theorem · functional analysis

ContinuousLinearMap.coprodEquiv_symm_apply

∀ {R : Type u_1} {S : Type u_2} {M : Type u_3} {M₁ : Type u_5} {M₂ : Type u_6} [inst : Semiring R]
  [inst_1 : TopologicalSpace M] [inst_2 : TopologicalSpace M₁] [inst_3 : TopologicalSpace M₂] [inst_4 : AddCommMonoid M]
  [inst_5 : Module R M] [inst_6 : ContinuousAdd M] [inst_7 : AddCommMonoid M₁] [inst_8 : Module R M₁]
  [inst_9 : AddCommMonoid M₂] [inst_10 : Module R M₂] [inst_11 : ContinuousAdd M₁] [inst_12 : ContinuousAdd M₂]
  [inst_13 : Semiring S] [inst_14 : Module S M] [inst_15 : ContinuousConstSMul S M] [inst_16 : SMulCommClass R S M]
  (f : M₁ × M₂ →L[R] M),
  ContinuousLinearMap.coprodEquiv.symm f =
    (f ∘SL ContinuousLinearMap.inl R M₁ M₂, f ∘SL ContinuousLinearMap.inr R M₁ M₂)

Defined in

Mathlib.Topology.Algebra.Module.ContinuousLinearMap.PiProd

Cited by

0 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringTopologicalSpaceTopologicalSpaceTopologicalSpaceAddCommMonoidModuleContinuousAddAddCommMonoidModuleAddCommMonoidModuleContinuousAddContinuousAddSemiringModuleContinuousConstSMulSMulCommClass