Theorems · Definition · functional analysis

ContDiffMapSupportedIn.monoCLM

(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] →
                    {n₁ n₂ : ℕ∞} →
                      {K₁ K₂ : TopologicalSpace.Compacts E} →
                        ContDiffMapSupportedIn E F n₁ K₁ →L[𝕜] ContDiffMapSupportedIn E F n₂ K₂

If n₁ ≥ n₂ and K₁ ⊆ K₂, monoCLM 𝕜 is the continuous 𝕜-linear inclusion of 𝓓^{n₁}_{K₁}(E, F) inside 𝓓^{n₂}_{K₂}(E, F). Otherwise, this is the zero map. Furthermore: * it is a topological embedding when n₁ = n₂ and K₁ ⊆ K₂ (not in Mathlib as of March 2026). * it maps bounded sets to compact sets when n₁ ≥ n₂ + 1 and K₁ ⊆ K₂ (not in Mathlib as of March 2026). The parameters n₁, n₂, K₁, K₂ are implicit as they can often be inferred from context, or specified by a type ascription.

Defined in

Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Cited by

4 results in Mathlib

Foundations

Depth 228 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClass