Theorems · Definition · geometry

ContinuousAffineMap.decompEquiv

(R : Type u_1) →
  (V : Type u_3) →
    {W : Type u_4} →
      (Q : Type u_5) →
        [inst : Ring R] →
          [inst_1 : AddCommGroup V] →
            [inst_2 : Module R V] →
              [inst_3 : TopologicalSpace V] →
                [IsTopologicalAddGroup V] →
                  [inst_5 : AddCommGroup W] →
                    [inst_6 : Module R W] →
                      [inst_7 : TopologicalSpace W] →
                        [inst_8 : AddTorsor W Q] →
                          [inst_9 : TopologicalSpace Q] → [IsTopologicalAddTorsor Q] → (V →ᴬ[R] Q) ≃ Q × (V →L[R] W)

The space of continuous affine maps from a topological vector space to a topological affine space is in bijection with 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

6 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupAddCommGroupModuleTopologicalSpaceAddTorsorTopologicalSpaceIsTopologicalAddTorsor