Theorems · Definition · functional analysis

ContinuousMultilinearMap.toFormalMultilinearSeries

{𝕜 : Type u} →
  {F : Type w} →
    [inst : Semiring 𝕜] →
      [inst_1 : AddCommMonoid F] →
        [inst_2 : Module 𝕜 F] →
          [inst_3 : TopologicalSpace F] →
            [inst_4 : ContinuousAdd F] →
              [inst_5 : ContinuousConstSMul 𝕜 F] →
                {ι : Type u_1} →
                  {E : ι → Type u_2} →
                    [inst_6 : (i : ι) → AddCommGroup (E i)] →
                      [inst_7 : (i : ι) → Module 𝕜 (E i)] →
                        [inst_8 : (i : ι) → TopologicalSpace (E i)] →
                          [inst_9 : ∀ (i : ι), IsTopologicalAddGroup (E i)] →
                            [inst_10 : ∀ (i : ι), ContinuousConstSMul 𝕜 (E i)] →
                              [Fintype ι] → ContinuousMultilinearMap 𝕜 E F → FormalMultilinearSeries 𝕜 ((i : ι) → E i) F

Realize a ContinuousMultilinearMap on ∀ i : ι, E i as the evaluation of a FormalMultilinearSeries by choosing an arbitrary identification ι ≃ Fin (Fintype.card ι).

Defined in

Mathlib.Analysis.Calculus.FormalMultilinearSeries

Cited by

5 results in Mathlib

Foundations

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

Assumes

SemiringAddCommMonoidModuleTopologicalSpaceContinuousAddContinuousConstSMulAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupContinuousConstSMulFintype