Theorems · Definition · functional analysis

InnerProductSpace.rankOne

(𝕜 : Type u_4) →
  {E : Type u_5} →
    {F : Type u_6} →
      [inst : RCLike 𝕜] →
        [inst_1 : SeminormedAddCommGroup E] →
          [inst_2 : NormedSpace 𝕜 E] →
            [inst_3 : SeminormedAddCommGroup F] → [inst_4 : InnerProductSpace 𝕜 F] → E →L[𝕜] F →L⋆[𝕜] F →L[𝕜] E

A rank-one operator on an inner product space is given by x ↦ y ↦ z ↦ ⟪y, z⟫ • x. This is also sometimes referred to as an outer product of vectors on a Hilbert space. This corresponds to the matrix outer product Matrix.vecMulVec, see InnerProductSpace.toMatrix_rankOne and InnerProductSpace.symm_toEuclideanLin_rankOne in Mathlib/Analysis/InnerProductSpace/PiL2.lean.

Defined in

Mathlib.Analysis.InnerProductSpace.LinearMap

Cited by

35 results in Mathlib

Foundations

Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupInnerProductSpace