Theorems · Definition · functional analysis

SchwartzMap.compCLM

(𝕜 : Type u_2) →
  {D : Type u_4} →
    {E : Type u_5} →
      {F : Type u_6} →
        [inst : NormedAddCommGroup E] →
          [inst_1 : NormedSpace ℝ E] →
            [inst_2 : NormedAddCommGroup F] →
              [inst_3 : NormedSpace ℝ F] →
                [inst_4 : RCLike 𝕜] →
                  [inst_5 : NormedAddCommGroup D] →
                    [inst_6 : NormedSpace ℝ D] →
                      [inst_7 : NormedSpace 𝕜 F] →
                        {g : D → E} →
                          Function.HasTemperateGrowth g →
                            (∃ k C, ∀ (x : D), ‖x‖ ≤ C * (1 + ‖g x‖) ^ k) → SchwartzMap E F →L[𝕜] SchwartzMap D F

Composition with a function on the right is a continuous linear map on Schwartz space provided that the function is temperate and growths polynomially near infinity.

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Cited by

1 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceRCLikeNormedAddCommGroupNormedSpaceNormedSpace