Theorems · Theorem · functional analysis

linearIndependent_of_ne_zero_of_inner_eq_zero

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {ι : Type u_4} {v : ι → E},
  (∀ (i : ι), v i ≠ 0) → (Pairwise fun i j => inner 𝕜 (v i) (v j) = 0) → LinearIndependent 𝕜 v

A family of vectors is linearly independent if they are nonzero and orthogonal.

Defined in

Mathlib.Analysis.InnerProductSpace.Basic

Cited by

3 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpace