Theorems · Theorem · functional analysis

RKHS.tendstoUniformlyOn_of_norm_kerFun_le

∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {X : Type u_2} {V : Type u_3} [inst_1 : NormedAddCommGroup V]
  [inst_2 : InnerProductSpace 𝕜 V] {H : Type u_4} [inst_3 : NormedAddCommGroup H] [inst_4 : InnerProductSpace 𝕜 H]
  [inst_5 : RKHS 𝕜 H X V] [inst_6 : CompleteSpace H] [inst_7 : CompleteSpace V] {C : ℝ} {s : Set X},
  (∀ x ∈ s, ‖RKHS.kerFun H x‖ ≤ C) →
    ∀ {ι : Type u_5} {l : Filter ι} {F : ι → H} {f : H},
      Filter.Tendsto F l (nhds f) → TendstoUniformlyOn (fun n => ⇑(F n)) (⇑f) l s

If the kernel functions are uniformly bounded on a set s (‖kerFun H x‖ ≤ C for x ∈ s), then convergence in H-norm implies uniform convergence of the underlying functions on s.

Defined in

Mathlib.Analysis.InnerProductSpace.Reproducing

Cited by

1 results in Mathlib

Foundations

Depth 199 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceRKHSCompleteSpaceCompleteSpace