Theorems · Definition · functional analysis

ContinuousLinearMap.coprodEquiv

{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] →
                                [ContinuousAdd M₁] →
                                  [ContinuousAdd M₂] →
                                    [inst_13 : Semiring S] →
                                      [inst_14 : Module S M] →
                                        [inst_15 : ContinuousConstSMul S M] →
                                          [inst_16 : SMulCommClass R S M] →
                                            ((M₁ →L[R] M) × (M₂ →L[R] M)) ≃ₗ[S] M₁ × M₂ →L[R] M

Taking the product of two maps with the same codomain is equivalent to taking the product of their domains. See note [bundled maps over different rings] for why separate R and S semirings are used. See ContinuousLinearMap.coprodEquivL for the ContinuousLinearEquiv version.

Defined in

Mathlib.Topology.Algebra.Module.ContinuousLinearMap.PiProd

Cited by

2 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringTopologicalSpaceTopologicalSpaceTopologicalSpaceAddCommMonoidModuleContinuousAddAddCommMonoidModuleAddCommMonoidModuleContinuousAddContinuousAddSemiringModuleContinuousConstSMulSMulCommClass