Theorems · Definition · functional analysis

SchwartzMap.bilinLeftCLM

{𝕜 : Type u_2} →
  {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 : NontriviallyNormedField 𝕜] →
                    [inst_5 : NormedAlgebra ℝ 𝕜] →
                      [inst_6 : NormedAddCommGroup D] →
                        [inst_7 : NormedSpace ℝ D] →
                          [inst_8 : NormedAddCommGroup G] →
                            [inst_9 : NormedSpace ℝ G] →
                              [inst_10 : NormedSpace 𝕜 F] →
                                [inst_11 : NormedSpace 𝕜 E] →
                                  [inst_12 : NormedSpace 𝕜 G] →
                                    (E →L[𝕜] F →L[𝕜] G) →
                                      {g : D → F} →
                                        Function.HasTemperateGrowth g → SchwartzMap D E →L[𝕜] SchwartzMap D G

The map f ↦ (x ↦ B (f x) (g x)) as a continuous 𝕜-linear map on Schwartz space, where B is a continuous 𝕜-linear map and g is a function of temperate growth.

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Cited by

3 results in Mathlib

Foundations

Depth 225 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNontriviallyNormedFieldNormedAlgebraNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedSpaceNormedSpaceNormedSpace