Theorems · Definition · functional analysis

OrthonormalBasis.fromOrthogonalSpanSingleton

{𝕜 : Type u_3} →
  [inst : RCLike 𝕜] →
    {E : Type u_4} →
      [inst_1 : NormedAddCommGroup E] →
        [inst_2 : InnerProductSpace 𝕜 E] →
          (n : ℕ) → [Fact (Module.finrank 𝕜 E = n + 1)] → {v : E} → v ≠ 0 → OrthonormalBasis (Fin n) 𝕜 ↥(𝕜 ∙ v)ᗮ

Given a natural number n one less than the finrank of a finite-dimensional inner product space, there exists an isometry from the orthogonal complement of a nonzero singleton to EuclideanSpace 𝕜 (Fin n).

Defined in

Mathlib.Analysis.InnerProductSpace.PiL2

Cited by

8 results in Mathlib

Foundations

Depth 241 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFact