Theorems · Definition · functional analysis

SchwartzMap.mkCLM

{𝕜 : Type u_2} →
  {𝕜' : Type u_3} →
    {D : Type u_4} →
      {E : Type u_5} →
        {F : Type u_6} →
          {G : Type u_7} →
            [inst : NormedAddCommGroup E] →
              [inst_1 : NormedSpace ℝ E] →
                [inst_2 : NormedAddCommGroup F] →
                  [inst_3 : NormedSpace ℝ F] →
                    [inst_4 : NormedField 𝕜] →
                      [inst_5 : NormedField 𝕜'] →
                        [inst_6 : NormedAddCommGroup D] →
                          [inst_7 : NormedSpace ℝ D] →
                            [inst_8 : NormedSpace 𝕜 E] →
                              [inst_9 : SMulCommClass ℝ 𝕜 E] →
                                [inst_10 : NormedAddCommGroup G] →
                                  [inst_11 : NormedSpace ℝ G] →
                                    [inst_12 : NormedSpace 𝕜' G] →
                                      [inst_13 : SMulCommClass ℝ 𝕜' G] →
                                        {σ : 𝕜 →+* 𝕜'} →
                                          [RingHomIsometric σ] →
                                            (A : SchwartzMap D E → F → G) →
                                              (∀ (f g : SchwartzMap D E) (x : F), A (f + g) x = A f x + A g x) →
                                                (∀ (a : 𝕜) (f : SchwartzMap D E) (x : F), A (a • f) x = σ a • A f x) →
                                                  (∀ (f : SchwartzMap D E), ContDiff ℝ (↑⊤) (A f)) →
                                                    (∀ (n : ℕ × ℕ),
                                                        ∃ s C,
                                                          0 ≤ C ∧
                                                            ∀ (f : SchwartzMap D E) (x : F),
                                                              ‖x‖ ^ n.1 * ‖iteratedFDeriv ℝ n.2 (A f) x‖ ≤
                                                                C * (s.sup (schwartzSeminormFamily 𝕜 D E)) f) →
                                                      SchwartzMap D E →SL[σ] SchwartzMap F G

Create a continuous semilinear map between Schwartz spaces. For an example of using this definition, see fderivCLM.

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Cited by

0 results in Mathlib

Foundations

Depth 224 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedFieldNormedFieldNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClassNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClassRingHomIsometric