Theorems · Definition · functional analysis

ContDiffMapSupportedIn.fderivLM

(𝕜 : Type u_1) →
  {E : Type u_2} →
    {F : Type u_3} →
      [inst : NontriviallyNormedField 𝕜] →
        [inst_1 : NormedAddCommGroup E] →
          [inst_2 : NormedSpace ℝ E] →
            [inst_3 : NormedAddCommGroup F] →
              [inst_4 : NormedSpace ℝ F] →
                [inst_5 : NormedSpace 𝕜 F] →
                  [inst_6 : SMulCommClass ℝ 𝕜 F] →
                    (n k : ℕ∞) →
                      {K : TopologicalSpace.Compacts E} →
                        ContDiffMapSupportedIn E F n K →ₗ[𝕜] ContDiffMapSupportedIn E (E →L[ℝ] F) k K

fderivLM 𝕜 n k is the 𝕜-linear-map sending f : 𝓓^{n}_{K}(E, F) to its derivative as an element of 𝓓^{k}_{K}(E, E →L[ℝ] F). This only makes mathematical sense if k + 1 ≤ n, otherwise we define it as the zero map. This is subsumed by fderivCLM, which also bundles the continuity.

Defined in

Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Cited by

6 results in Mathlib

Foundations

Depth 216 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClass