Theorems · Theorem · functional analysis

ContinuousLinearMap.norm_eq_iInf_range_iff_adjoint_apply_eq_zero

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : InnerProductSpace 𝕜 E] [inst_4 : InnerProductSpace 𝕜 F]
  [inst_5 : CompleteSpace E] [inst_6 : CompleteSpace F] (A : E →L[𝕜] F) (y : F) (x : E),
  ‖y - A x‖ = ⨅ z, ‖y - ↑z‖ ↔ (ContinuousLinearMap.adjoint A) (y - A x) = 0

The fitted value A x minimizes the distance to y among points in A.range if and only if the adjoint of A sends the residual y - A x to zero.

Defined in

Mathlib.Analysis.InnerProductSpace.Adjoint

Cited by

1 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedAddCommGroupInnerProductSpaceInnerProductSpaceCompleteSpaceCompleteSpace