Theorems · Theorem · functional analysis

Orthonormal.inner_left_sum

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

The inner product of a linear combination of a set of orthonormal vectors with one of those vectors picks out the coefficient of that vector.

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