Theorems · Theorem · functional analysis

ContinuousLinearEquiv.norm_symm_pos

∀ {𝕜 : Type u_1} {𝕜₂ : Type u_3} {E : Type u_5} {F : Type u_6} [inst : NormedAddCommGroup E]
  [inst_1 : NormedAddCommGroup F] [inst_2 : NontriviallyNormedField 𝕜] [inst_3 : NontriviallyNormedField 𝕜₂]
  [inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜}
  [inst_6 : RingHomInvPair σ₁₂ σ₂₁] [inst_7 : RingHomInvPair σ₂₁ σ₁₂] [RingHomIsometric σ₂₁] [RingHomIsometric σ₁₂]
  [Nontrivial E] (e : E ≃SL[σ₁₂] F), 0 < ‖↑e.symm‖

Defined in

Mathlib.Analysis.Normed.Operator.NormedSpace

Cited by

2 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceRingHomInvPairRingHomInvPairRingHomIsometricRingHomIsometricNontrivial