Theorems · Definition · functional analysis

FormalMultilinearSeries.compContinuousLinearMap

{𝕜 : Type u} →
  {E : Type v} →
    {F : Type w} →
      {G : Type x} →
        [inst : Semiring 𝕜] →
          [inst_1 : AddCommMonoid E] →
            [inst_2 : Module 𝕜 E] →
              [inst_3 : TopologicalSpace E] →
                [inst_4 : ContinuousAdd E] →
                  [inst_5 : ContinuousConstSMul 𝕜 E] →
                    [inst_6 : AddCommMonoid F] →
                      [inst_7 : Module 𝕜 F] →
                        [inst_8 : TopologicalSpace F] →
                          [inst_9 : ContinuousAdd F] →
                            [inst_10 : ContinuousConstSMul 𝕜 F] →
                              [inst_11 : AddCommMonoid G] →
                                [inst_12 : Module 𝕜 G] →
                                  [inst_13 : TopologicalSpace G] →
                                    [inst_14 : ContinuousAdd G] →
                                      [inst_15 : ContinuousConstSMul 𝕜 G] →
                                        FormalMultilinearSeries 𝕜 F G → (E →L[𝕜] F) → FormalMultilinearSeries 𝕜 E G

Composing each term pₙ in a formal multilinear series with (u, ..., u) where u is a fixed continuous linear map, gives a new formal multilinear series p.compContinuousLinearMap u.

Defined in

Mathlib.Analysis.Calculus.FormalMultilinearSeries

Cited by

35 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModuleTopologicalSpaceContinuousAddContinuousConstSMulAddCommMonoidModuleTopologicalSpaceContinuousAddContinuousConstSMulAddCommMonoidModuleTopologicalSpaceContinuousAddContinuousConstSMul