Theorems · Theorem · functional analysis

maximal_orthonormal_iff_orthogonalComplement_eq_bot

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {v : Set E}, Orthonormal 𝕜 Subtype.val → ((∀ u ⊇ v, Orthonormal 𝕜 Subtype.val → u = v) ↔ (Submodule.span 𝕜 v)ᗮ = ⊥)

An orthonormal set in an InnerProductSpace is maximal, if and only if the orthogonal complement of its span is empty.

Defined in

Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Cited by

3 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpace