Theorems · Theorem · functional analysis

RKHS.kerFun_dense

∀ {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],
  (Submodule.span 𝕜 {x | ∃ x_1 v, (RKHS.kerFun H x_1) v = x}).topologicalClosure = ⊤

The span of the kernel functions is dense.

Defined in

Mathlib.Analysis.InnerProductSpace.Reproducing

Cited by

0 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceRKHSCompleteSpaceCompleteSpace