Theorems · Definition · several complex variables

FormalMultilinearSeries.comp

{𝕜 : Type u_1} →
  {E : Type u_2} →
    {F : Type u_3} →
      {G : Type u_4} →
        [inst : CommRing 𝕜] →
          [inst_1 : AddCommGroup E] →
            [inst_2 : AddCommGroup F] →
              [inst_3 : AddCommGroup G] →
                [inst_4 : Module 𝕜 E] →
                  [inst_5 : Module 𝕜 F] →
                    [inst_6 : Module 𝕜 G] →
                      [inst_7 : TopologicalSpace E] →
                        [inst_8 : TopologicalSpace F] →
                          [inst_9 : TopologicalSpace G] →
                            [inst_10 : IsTopologicalAddGroup E] →
                              [inst_11 : ContinuousConstSMul 𝕜 E] →
                                [inst_12 : IsTopologicalAddGroup F] →
                                  [inst_13 : ContinuousConstSMul 𝕜 F] →
                                    [inst_14 : IsTopologicalAddGroup G] →
                                      [inst_15 : ContinuousConstSMul 𝕜 G] →
                                        FormalMultilinearSeries 𝕜 F G →
                                          FormalMultilinearSeries 𝕜 E F → FormalMultilinearSeries 𝕜 E G

Formal composition of two formal multilinear series. The n-th coefficient in the composition is defined to be the sum of q.compAlongComposition p c over all compositions of n. In other words, this term (as a multilinear function applied to v_0, ..., v_{n-1}) is ∑'_{k} ∑'_{i₁ + ... + iₖ = n} qₖ (p_{i_1} (...), ..., p_{i_k} (...)), where one puts all variables v_0, ..., v_{n-1} in increasing order in the dots. In general, the composition q ∘ p only makes sense when the constant coefficient of p vanishes. We give a general formula but which ignores the value of p 0 instead.

Defined in

Mathlib.Analysis.Analytic.Composition

Cited by

25 results in Mathlib

Foundations

Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupAddCommGroupAddCommGroupModuleModuleModuleTopologicalSpaceTopologicalSpaceTopologicalSpaceIsTopologicalAddGroupContinuousConstSMulIsTopologicalAddGroupContinuousConstSMulIsTopologicalAddGroupContinuousConstSMul