Theorems · Theorem · functional analysis

ContinuousLinearMap.opNorm_ext

∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8}
  [inst : SeminormedAddCommGroup E] [inst_1 : SeminormedAddCommGroup F] [inst_2 : SeminormedAddCommGroup G]
  [inst_3 : NontriviallyNormedField 𝕜] [inst_4 : NontriviallyNormedField 𝕜₂] [inst_5 : NontriviallyNormedField 𝕜₃]
  [inst_6 : NormedSpace 𝕜 E] [inst_7 : NormedSpace 𝕜₂ F] [inst_8 : NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₁₃ : 𝕜 →+* 𝕜₃}
  [RingHomIsometric σ₁₃] (f : E →SL[σ₁₂] F) (g : E →SL[σ₁₃] G), (∀ (x : E), ‖f x‖ = ‖g x‖) → ‖f‖ = ‖g‖

Defined in

Mathlib.Analysis.Normed.Operator.Bilinear

Cited by

1 results in Mathlib

Foundations

Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroupSeminormedAddCommGroupSeminormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceNormedSpaceRingHomIsometric