Theorems · Theorem · functional analysis

Orthonormal.sum_inner_products_le

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {ι : Type u_4} (x : E) {v : ι → E} {s : Finset ι}, Orthonormal 𝕜 v → ∑ i ∈ s, ‖inner 𝕜 (v i) x‖ ^ 2 ≤ ‖x‖ ^ 2

Bessel's inequality for finite sums.

Defined in

Mathlib.Analysis.InnerProductSpace.Orthonormal

Cited by

2 results in Mathlib

Foundations

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

Assumes

RCLikeSeminormedAddCommGroupInnerProductSpace