Theorems · Definition · functional analysis

FiniteDimensional.orthonormalBasisSingleton

(ι : Type u_1) →
  (𝕜 : Type u_3) →
    [inst : RCLike 𝕜] →
      {E : Type u_4} →
        [inst_1 : NormedAddCommGroup E] →
          [inst_2 : InnerProductSpace 𝕜 E] →
            [inst_3 : Fintype ι] → [Unique ι] → Module.finrank 𝕜 E = 1 → (v : E) → ‖v‖ = 1 → OrthonormalBasis ι 𝕜 E

In an inner product space with dimension 1, a set {v} is an orthonormal basis for ‖v‖ = 1.

Defined in

Mathlib.Analysis.InnerProductSpace.PiL2

Cited by

5 results in Mathlib

Foundations

Depth 229 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFintypeUnique