Theorems · Theorem · functional analysis

OrthogonalFamily.eq_ite

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {ι : Type u_4} {G : ι → Type u_5} [inst_3 : (i : ι) → NormedAddCommGroup (G i)]
  [inst_4 : (i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E},
  OrthogonalFamily 𝕜 G V →
    ∀ [inst_5 : DecidableEq ι] {i j : ι} (v : G i) (w : G j),
      inner 𝕜 ((V i) v) ((V j) w) = if i = j then inner 𝕜 ((V i) v) ((V j) w) else 0

Defined in

Mathlib.Analysis.InnerProductSpace.Subspace

Cited by

3 results in Mathlib

Foundations

Depth 47 from the axioms · uses propext, Quot.sound

Assumes

RCLikeSeminormedAddCommGroupInnerProductSpaceNormedAddCommGroupInnerProductSpaceDecidableEq