Theorems · Definition · functional analysis

ContDiffMapSupportedIn.integralAgainstBilinCLM

{𝕜 : Type u_1} →
  {E : Type u_2} →
    [inst : NontriviallyNormedField 𝕜] →
      [inst_1 : NormedAddCommGroup E] →
        [inst_2 : NormedSpace ℝ E] →
          {n : ℕ∞} →
            {K : TopologicalSpace.Compacts E} →
              {m : MeasurableSpace E} →
                [OpensMeasurableSpace E] →
                  {F₁ : Type u_5} →
                    {F₂ : Type u_6} →
                      {F₃ : Type u_7} →
                        [inst_4 : NormedAddCommGroup F₁] →
                          [inst_5 : NormedSpace 𝕜 F₁] →
                            [inst_6 : NormedSpace ℝ F₁] →
                              [inst_7 : NormedAddCommGroup F₂] →
                                [inst_8 : NormedSpace 𝕜 F₂] →
                                  [inst_9 : NormedAddCommGroup F₃] →
                                    [inst_10 : NormedSpace 𝕜 F₃] →
                                      [inst_11 : SMulCommClass ℝ 𝕜 F₁] →
                                        [inst_12 : NormedSpace ℝ F₃] →
                                          [SMulCommClass ℝ 𝕜 F₃] →
                                            (F₁ →L[𝕜] F₂ →L[𝕜] F₃) →
                                              MeasureTheory.Measure E →
                                                (E → F₂) → ContDiffMapSupportedIn E F₁ n K →L[𝕜] F₃

Given a continuous 𝕜-bilinear map B : F₁ →L[𝕜] F₂ →L[𝕜] F₃, a measure μ on E, and a function φ : E → F₂ which is integrable on K, this is the continuous 𝕜-linear map f ↦ ∫ x, B (f x) (φ x) ∂μ from 𝓓^{n}_{K}(E, F₁) to F₃. Otherwise, this is the zero map.

Defined in

Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Cited by

5 results in Mathlib

Foundations

Depth 264 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceOpensMeasurableSpaceNormedAddCommGroupNormedSpaceNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceSMulCommClassNormedSpaceSMulCommClass