Theorems · Definition · functional analysis

TestFunction.lineDerivCLM

(𝕜 : Type u_1) →
  [inst : NontriviallyNormedField 𝕜] →
    {E : Type u_3} →
      [inst_1 : NormedAddCommGroup E] →
        [inst_2 : NormedSpace ℝ E] →
          {Ω : TopologicalSpace.Opens E} →
            {F : Type u_4} →
              [inst_3 : NormedAddCommGroup F] →
                [inst_4 : NormedSpace ℝ F] →
                  [inst_5 : NormedSpace 𝕜 F] →
                    {n k : ℕ∞} →
                      [inst_6 : Algebra ℝ 𝕜] →
                        [inst_7 : IsScalarTower ℝ 𝕜 F] → E → TestFunction Ω F n →L[𝕜] TestFunction Ω F k

lineDerivCLM 𝕜 v is the continuous 𝕜-linear-map sending f : 𝓓^{n}_{K}(E, F) to its derivative along the vector v, which is an element of 𝓓^{k}_{K}(E, F). This only makes mathematical sense if k + 1 ≤ n, otherwise we define it as the zero map. The parameters n and k are implicit as they can often be inferred from context, or specified by a type ascription. For n = k = ⊤, we also provide instances of the LineDeriv notation typeclass.

Defined in

Mathlib.Analysis.Distribution.TestFunction

Cited by

12 results in Mathlib

Foundations

Depth 231 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedSpaceAlgebraIsScalarTower