Theorems · Theorem · functional analysis

OrthonormalBasis.prod_apply

∀ {𝕜 : Type u_1} {ι₁ : Type u_2} {ι₂ : Type u_3} {E : Type u_4} {F : Type u_5} [inst : RCLike 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E] [inst_3 : NormedAddCommGroup F]
  [inst_4 : InnerProductSpace 𝕜 F] [inst_5 : Fintype ι₁] [inst_6 : Fintype ι₂] (v : OrthonormalBasis ι₁ 𝕜 E)
  (w : OrthonormalBasis ι₂ 𝕜 F) (i : ι₁ ⊕ ι₂),
  (v.prod w) i = Sum.elim (WithLp.toLp 2 ∘ ⇑(LinearMap.inl 𝕜 E F) ∘ ⇑v) (WithLp.toLp 2 ∘ ⇑(LinearMap.inr 𝕜 E F) ∘ ⇑w) i

Defined in

Mathlib.Analysis.InnerProductSpace.ProdL2

Cited by

0 results in Mathlib

Foundations

Depth 233 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceFintypeFintype