Theorems · Theorem · functional analysis

SchwartzMap.one_add_le_sup_seminorm_apply

∀ {𝕜 : Type u_2} {E : Type u_5} {F : Type u_6} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] [inst_4 : NormedField 𝕜] [inst_5 : NormedSpace 𝕜 F]
  [inst_6 : SMulCommClass ℝ 𝕜 F] {m : ℕ × ℕ} {k n : ℕ},
  k ≤ m.1 →
    n ≤ m.2 →
      ∀ (f : SchwartzMap E F) (x : E),
        (1 + ‖x‖) ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤
          2 ^ m.1 * ((Finset.Iic m).sup fun m => SchwartzMap.seminorm 𝕜 m.1 m.2) f

A more convenient version of le_sup_seminorm_apply. The set Finset.Iic m is the set of all pairs (k', n') with k' ≤ m.1 and n' ≤ m.2. Note that the constant is far from optimal.

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Basic

Cited by

1 results in Mathlib

Foundations

Depth 219 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedFieldNormedSpaceSMulCommClass