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Theorems Β· Definition Β· functional analysis

ContDiffMapSupportedIn.mkCLM

(π•œ : Type u_1) β†’
  {E : Type u_2} β†’
    {F : Type u_3} β†’
      {F' : Type u_4} β†’
        [inst : NontriviallyNormedField π•œ] β†’
          [inst_1 : NormedAddCommGroup E] β†’
            [inst_2 : NormedSpace ℝ E] β†’
              [inst_3 : NormedAddCommGroup F] β†’
                [inst_4 : NormedSpace ℝ F] β†’
                  [inst_5 : NormedSpace π•œ F] β†’
                    [inst_6 : SMulCommClass ℝ π•œ F] β†’
                      [inst_7 : NormedAddCommGroup F'] β†’
                        [inst_8 : NormedSpace ℝ F'] β†’
                          [inst_9 : NormedSpace π•œ F'] β†’
                            [inst_10 : SMulCommClass ℝ π•œ F'] β†’
                              {n₁ nβ‚‚ : β„•βˆž} β†’
                                {K₁ Kβ‚‚ : TopologicalSpace.Compacts E} β†’
                                  (A : ContDiffMapSupportedIn E F n₁ K₁ β†’ E β†’ F') β†’
                                    (βˆ€ (f g : ContDiffMapSupportedIn E F n₁ K₁) (x : E), A (f + g) x = A f x + A g x) β†’
                                      (βˆ€ (c : π•œ) (f : ContDiffMapSupportedIn E F n₁ K₁) (x : E),
                                          A (c β€’ f) x = c β€’ A f x) β†’
                                        (βˆ€ (f : ContDiffMapSupportedIn E F n₁ K₁), ContDiff ℝ (↑nβ‚‚) (A f)) β†’
                                          (βˆ€ (f : ContDiffMapSupportedIn E F n₁ K₁), Set.EqOn (A f) 0 (↑Kβ‚‚)ᢜ) β†’
                                            (βˆ€ (i : β„•),
                                                ↑i ≀ nβ‚‚ β†’
                                                  βˆƒ s C,
                                                    0 ≀ C ∧
                                                      βˆ€ (f : ContDiffMapSupportedIn E F n₁ K₁),
                                                        βˆ€ x ∈ Kβ‚‚,
                                                          β€–iteratedFDeriv ℝ i (A f) xβ€– ≀
                                                            C *
                                                              (s.sup fun j =>
                                                                  ContDiffMapSupportedIn.seminorm π•œ E F n₁ K₁ j)
                                                                f) β†’
                                              ContDiffMapSupportedIn E F n₁ K₁ β†’L[π•œ] ContDiffMapSupportedIn E F' nβ‚‚ Kβ‚‚

Define a continuous π•œ-linear map from 𝓓^{n₁}_{K₁}(E, F) to 𝓓^{nβ‚‚}_{Kβ‚‚}(E, F').

Defined in
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn
Cited by
0 results in Mathlib
Foundations
Depth 226 from the axioms Β· uses propext, Classical.choice, Quot.sound
Assumes
NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClassNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClass
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

ContDiffMapSupportedIn.bilinLeftCLM Β· cited by 1ContDiffMapSupportedIn.bi…DFunLike.coe Β· cited by 62936DFunLike.coeSet Β· cited by 53352SetReal Β· cited by 25697RealRingHom.id Β· cited by 18349RingHom.idNormedAddCommGroup Β· cited by 15752NormedAddCommGroupFinset Β· cited by 13712FinsetNormedSpace Β· cited by 12499NormedSpaceNontriviallyNormedField Β· cited by 8742NontriviallyNormedFieldSetLike.coe Β· cited by 8199SetLike.coeNorm.norm Β· cited by 5413Norm.normContinuousLinearMap Β· cited by 5352ContinuousLinearMapENat Β· cited by 4985ENatCompl.compl Β· cited by 2925Compl.complSMulCommClass Β· cited by 1927SMulCommClassWithTop.some Β· cited by 1128WithTop.someContDiffMapSupportedIn.mkCLMCITED BYCITES

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