Theorems · Theorem · functional analysis

RKHS.kernel_inner

∀ {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] (x y : X) (v w : V),
  inner 𝕜 ((RKHS.kernel H x y) v) w = inner 𝕜 ((RKHS.kerFun H y) v) ((RKHS.kerFun H x) w)

The "reproducing" property of the kernel.

Defined in

Mathlib.Analysis.InnerProductSpace.Reproducing

Cited by

2 results in Mathlib

Foundations

Depth 194 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceRKHSCompleteSpaceCompleteSpace