Theorems · Theorem · functional analysis

ContDiffMapSupportedIn.continuous_iff_comp

∀ {E : Type u_2} {F : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F]
  [inst_3 : NormedSpace ℝ F] {n : ℕ∞} {K : TopologicalSpace.Compacts E} {X : Type u_5} [inst_4 : TopologicalSpace X]
  (φ : X → ContDiffMapSupportedIn E F n K),
  Continuous φ ↔ ∀ (i : ℕ), Continuous (⇑(ContDiffMapSupportedIn.structureMapCLM ℝ n i) ∘ φ)

The universal property of the topology on 𝓓^{n}_{K}(E, F): a map to 𝓓^{n}_{K}(E, F) is continuous if and only if its composition with each structure map structureMapCLM ℝ n i : 𝓓^{n}_{K}(E, F) → (E →ᵇ (E [×i]→L[ℝ] F)) is continuous. Since structureMapCLM ℝ n i is zero whenever i > n, it suffices to check it for i ≤ n, as proven by continuous_iff_comp_order_le.

Defined in

Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Cited by

1 results in Mathlib

Foundations

Depth 221 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpace