Theorems · Theorem · functional analysis

LinearIsometry.orthonormal_comp_iff

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {ι : Type u_4} {E' : Type u_7} [inst_3 : SeminormedAddCommGroup E'] [inst_4 : InnerProductSpace 𝕜 E'] {v : ι → E}
  (f : E →ₗᵢ[𝕜] E'), Orthonormal 𝕜 (⇑f ∘ v) ↔ Orthonormal 𝕜 v

A linear isometry preserves the property of being orthonormal.

Defined in

Mathlib.Analysis.InnerProductSpace.Orthonormal

Cited by

2 results in Mathlib

Foundations

Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeSeminormedAddCommGroupInnerProductSpaceSeminormedAddCommGroupInnerProductSpace