Theorems · Theorem · operator theory

ContinuousLinearMap.IsIdempotentElem.isPositive_iff_isSelfAdjoint

∀ {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.IsPositive ↔ IsSelfAdjoint p)

An idempotent operator is positive if and only if it is self-adjoint.

Defined in

Mathlib.Analysis.InnerProductSpace.Positive

Cited by

2 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpace