Theorems · Theorem · operator theory

ContinuousLinearMap.IsIdempotentElem.TFAE

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  [inst_3 : CompleteSpace E] {p : E →L[𝕜] E},
  IsIdempotentElem p → [(↑p).rangeᗮ = (↑p).ker, IsStarNormal p, IsSelfAdjoint p, p.IsPositive].TFAE

For an idempotent operator p, TFAE: * (range p)ᗮ = ker p * p is normal * p is self-adjoint * p is positive

Defined in

Mathlib.Analysis.InnerProductSpace.Positive

Cited by

0 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpace