Theorems · Theorem · operator theory

ContinuousLinearMap.isPositive_iff_eq_sum_rankOne

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  [FiniteDimensional 𝕜 E] {T : E →L[𝕜] E}, T.IsPositive ↔ ∃ m u, T = ∑ i, ((InnerProductSpace.rankOne 𝕜) (u i)) (u i)

In finite-dimensional spaces, a continuous linear map is positive iff it is equal to the sum of rank-one positive operators.

Defined in

Mathlib.Analysis.InnerProductSpace.Positive

Cited by

1 results in Mathlib

Foundations

Depth 259 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFiniteDimensional