Theorems · Theorem · functional analysis

Module.Basis.ext_linearIsometryEquiv

∀ {R : Type u_1} {R₂ : Type u_2} {E : Type u_5} {E₂ : Type u_6} [inst : Semiring R] [inst_1 : Semiring R₂]
  {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [inst_2 : RingHomInvPair σ₁₂ σ₂₁] [inst_3 : RingHomInvPair σ₂₁ σ₁₂]
  [inst_4 : SeminormedAddCommGroup E] [inst_5 : SeminormedAddCommGroup E₂] [inst_6 : Module R E] [inst_7 : Module R₂ E₂]
  {ι : Type u_11} (b : Module.Basis ι R E) {f₁ f₂ : E ≃ₛₗᵢ[σ₁₂] E₂}, (∀ (i : ι), f₁ (b i) = f₂ (b i)) → f₁ = f₂

Two linear isometric equivalences are equal if they are equal on basis vectors.

Defined in

Mathlib.Analysis.Normed.Operator.LinearIsometry

Cited by

5 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringRingHomInvPairRingHomInvPairSeminormedAddCommGroupSeminormedAddCommGroupModuleModule