Theorems · Definition · geometry

ContinuousAffineMap.decompAffineEquiv

(R : Type u_1) →
  (S : Type u_2) →
    (V : Type u_3) →
      {W : Type u_4} →
        (Q : Type u_5) →
          [inst : Ring S] →
            [inst_1 : Ring R] →
              [inst_2 : AddCommGroup V] →
                [inst_3 : Module R V] →
                  [inst_4 : TopologicalSpace V] →
                    [IsTopologicalAddGroup V] →
                      [inst_6 : AddCommGroup W] →
                        [inst_7 : Module R W] →
                          [inst_8 : TopologicalSpace W] →
                            [inst_9 : Module S W] →
                              [inst_10 : SMulCommClass R S W] →
                                [inst_11 : ContinuousConstSMul S W] →
                                  [inst_12 : AddTorsor W Q] →
                                    [inst_13 : TopologicalSpace Q] →
                                      [inst_14 : IsTopologicalAddGroup W] →
                                        [inst_15 : IsTopologicalAddTorsor Q] → (V →ᴬ[R] Q) ≃ᵃ[S] Q × (V →L[R] W)

The space of continuous affine maps from a topological vector space to a topological affine space is affinely isomorphic to the product of the codomain with the space of linear maps, by taking the value of the affine map at (0 : V) and the linear part.

Defined in

Mathlib.Topology.Algebra.ContinuousAffineMap

Cited by

5 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingRingAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupAddCommGroupModuleTopologicalSpaceModuleSMulCommClassContinuousConstSMulAddTorsorTopologicalSpaceIsTopologicalAddGroupIsTopologicalAddTorsor