Theorems · Theorem · operator theory

LinearMap.IsSymmetric.inner_map_polarization

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {T : E →ₗ[𝕜] E},
  T.IsSymmetric →
    ∀ (x y : E),
      inner 𝕜 (T x) y =
        (inner 𝕜 (T (x + y)) (x + y) - inner 𝕜 (T (x - y)) (x - y) -
              RCLike.I * inner 𝕜 (T (x + RCLike.I • y)) (x + RCLike.I • y) +
            RCLike.I * inner 𝕜 (T (x - RCLike.I • y)) (x - RCLike.I • y)) /
          4

Polarization identity for symmetric linear maps. See inner_map_polarization for the complex version without the symmetric assumption.

Defined in

Mathlib.Analysis.InnerProductSpace.Symmetric

Cited by

1 results in Mathlib

Foundations

Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeSeminormedAddCommGroupInnerProductSpace