Theorems · Theorem · functional analysis

ContinuousLinearMap.inner_map_map_iff_adjoint_comp_self

∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {H : Type u_5} [inst_1 : NormedAddCommGroup H] [inst_2 : InnerProductSpace 𝕜 H]
  [inst_3 : CompleteSpace H] {K : Type u_6} [inst_4 : NormedAddCommGroup K] [inst_5 : InnerProductSpace 𝕜 K]
  [inst_6 : CompleteSpace K] (u : H →L[𝕜] K),
  (∀ (x y : H), inner 𝕜 (u x) (u y) = inner 𝕜 x y) ↔ ContinuousLinearMap.adjoint u ∘SL u = 1

Defined in

Mathlib.Analysis.InnerProductSpace.Adjoint

Cited by

2 results in Mathlib

Foundations

Depth 194 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpaceNormedAddCommGroupInnerProductSpaceCompleteSpace