Theorems · Theorem · functional analysis

maximal_orthonormal_iff_basis_of_finiteDimensional

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {v : Set E} [FiniteDimensional 𝕜 E],
  Orthonormal 𝕜 Subtype.val → ((∀ u ⊇ v, Orthonormal 𝕜 Subtype.val → u = v) ↔ ∃ b, ⇑b = Subtype.val)

An orthonormal set in a finite-dimensional InnerProductSpace is maximal, if and only if it is a basis.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Cited by

0 results in Mathlib

Foundations

Depth 182 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFiniteDimensional