Theorems · Theorem · functional analysis

Orthonormal.inner_left_right_finset

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {ι : Type u_4} {s : Finset ι} {v : ι → E},
  Orthonormal 𝕜 v → ∀ {a : ι → ι → 𝕜}, ∑ i ∈ s, ∑ j ∈ s, a i j • inner 𝕜 (v j) (v i) = ∑ k ∈ s, a k k

The double sum of weighted inner products of pairs of vectors from an orthonormal sequence is the sum of the weights.

Defined in

Mathlib.Analysis.InnerProductSpace.Orthonormal

Cited by

1 results in Mathlib

Foundations

Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeSeminormedAddCommGroupInnerProductSpace