Theorems · Theorem · functional analysis

ContinuousLinearMap.opNorm_comp_le

∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_5} {G : Type u_7}
  [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} {σ₂₃ : 𝕜₂ →+* 𝕜₃}
  {σ₁₃ : 𝕜 →+* 𝕜₃} [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₁₂] [RingHomIsometric σ₂₃]
  (h : F →SL[σ₂₃] G) (f : E →SL[σ₁₂] F), ‖h ∘SL f‖ ≤ ‖h‖ * ‖f‖

The operator norm is submultiplicative.

Defined in

Mathlib.Analysis.Normed.Operator.Basic

Cited by

16 results in Mathlib

Foundations

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

Assumes

SeminormedAddCommGroupSeminormedAddCommGroupSeminormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceNormedSpaceRingHomCompTripleRingHomIsometricRingHomIsometric