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Theorems Β· Definition Β· functional analysis

SchwartzMap.mkLM

{π•œ : Type u_2} β†’
  {π•œ' : Type u_3} β†’
    {D : Type u_4} β†’
      {E : Type u_5} β†’
        {F : Type u_6} β†’
          {G : Type u_7} β†’
            [inst : NormedAddCommGroup E] β†’
              [inst_1 : NormedSpace ℝ E] β†’
                [inst_2 : NormedAddCommGroup F] β†’
                  [inst_3 : NormedSpace ℝ F] β†’
                    [inst_4 : NormedField π•œ] β†’
                      [inst_5 : NormedField π•œ'] β†’
                        [inst_6 : NormedAddCommGroup D] β†’
                          [inst_7 : NormedSpace ℝ D] β†’
                            [inst_8 : NormedSpace π•œ E] β†’
                              [inst_9 : SMulCommClass ℝ π•œ E] β†’
                                [inst_10 : NormedAddCommGroup G] β†’
                                  [inst_11 : NormedSpace ℝ G] β†’
                                    [inst_12 : NormedSpace π•œ' G] β†’
                                      [inst_13 : SMulCommClass ℝ π•œ' G] β†’
                                        {Οƒ : π•œ β†’+* π•œ'} β†’
                                          (A : SchwartzMap D E β†’ F β†’ G) β†’
                                            (βˆ€ (f g : SchwartzMap D E) (x : F), A (f + g) x = A f x + A g x) β†’
                                              (βˆ€ (a : π•œ) (f : SchwartzMap D E) (x : F), A (a β€’ f) x = Οƒ a β€’ A f x) β†’
                                                (βˆ€ (f : SchwartzMap D E), ContDiff ℝ (β†‘βŠ€) (A f)) β†’
                                                  (βˆ€ (n : β„• Γ— β„•),
                                                      βˆƒ s C,
                                                        0 ≀ C ∧
                                                          βˆ€ (f : SchwartzMap D E) (x : F),
                                                            β€–xβ€– ^ n.1 * β€–iteratedFDeriv ℝ n.2 (A f) xβ€– ≀
                                                              C * (s.sup (schwartzSeminormFamily π•œ D E)) f) β†’
                                                    SchwartzMap D E β†’β‚›β‚—[Οƒ] SchwartzMap F G

Create a semilinear map between Schwartz spaces. Note: This is a helper definition for mkCLM.

Defined in
Mathlib.Analysis.Distribution.SchwartzSpace.Basic
Cited by
0 results in Mathlib
Foundations
Depth 222 from the axioms Β· uses propext, Classical.choice, Quot.sound
Assumes
NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedFieldNormedFieldNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClassNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClass
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Around this declaration

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

SchwartzMap.mkCLM Β· cited by 0SchwartzMap.mkCLMDFunLike.coe Β· cited by 62936DFunLike.coeReal Β· cited by 25697RealNormedAddCommGroup Β· cited by 15752NormedAddCommGroupFinset Β· cited by 13712FinsetNormedSpace Β· cited by 12499NormedSpaceLinearMap Β· cited by 10215LinearMapRingHom Β· cited by 10189RingHomTop.top Β· cited by 9680Top.topNorm.norm Β· cited by 5413Norm.normENat Β· cited by 4985ENatSMulCommClass Β· cited by 1927SMulCommClassWithTop.some Β· cited by 1128WithTop.someNormedField Β· cited by 1084NormedFieldContinuousMultilinearMap Β· cited by 1016ContinuousMultilinearMapFinset.sup Β· cited by 530Finset.supSchwartzMap.mkLMCITED BYCITES

Cites20

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

  • DFunLike.coestatement and proof Β· cited by 62,936
  • Realstatement and proof Β· cited by 25,697
  • NormedAddCommGroupstatement and proof Β· cited by 15,752
  • Finsetstatement and proof Β· cited by 13,712
  • NormedSpacestatement and proof Β· cited by 12,499
  • LinearMapstatement Β· cited by 10,215
  • RingHomstatement and proof Β· cited by 10,189
  • Top.topstatement and proof Β· cited by 9,680
  • Norm.normstatement and proof Β· cited by 5,413
  • ENatstatement Β· cited by 4,985
  • SMulCommClassstatement and proof Β· cited by 1,927
  • WithTop.somestatement and proof Β· cited by 1,128

Cited by1

Results whose statement or proof uses this declaration.