Theorems · Definition · functional analysis

ContDiffMapSupportedIn.structureMapLM

(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 : ℕ∞) →
                      {K : TopologicalSpace.Compacts E} →
                        (i : ℕ) → ContDiffMapSupportedIn E F n K →ₗ[𝕜] BoundedContinuousFunction E (E [×i]→L[ℝ] F)

structureMapLM 𝕜 n i is the 𝕜-linear-map sending f : 𝓓^{n}_{K}(E, F) to its i-th iterated derivative as an element of E →ᵇ (E [×i]→L[ℝ] F). In other words, it is the composition of toBoundedContinuousFunctionLM 𝕜 and iteratedFDerivLM 𝕜 n 0 i. This only makes mathematical sense if i ≤ n, otherwise we define it as the zero map. We call these "structure maps" because they define the topology on 𝓓^{n}_{K}(E, F). This is subsumed by structureMapCLM, which also bundles the continuity.

Defined in

Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Cited by

11 results in Mathlib

Foundations

Depth 218 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClass