Theorems · Theorem · functional analysis

Orthonormal.inner_finsupp_eq_sum_left

∀ {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 →
    ∀ (l₁ l₂ : ι →₀ 𝕜),
      inner 𝕜 ((Finsupp.linearCombination 𝕜 v) l₁) ((Finsupp.linearCombination 𝕜 v) l₂) =
        l₁.sum fun i y => (starRingEnd 𝕜) y * l₂ i

The inner product of two linear combinations of a set of orthonormal vectors, expressed as a sum over the first Finsupp.

Defined in

Mathlib.Analysis.InnerProductSpace.Orthonormal

Cited by

0 results in Mathlib

Foundations

Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeSeminormedAddCommGroupInnerProductSpace