Theorems · Theorem · functional analysis

ContinuousLinearMap.le_opNorm

∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [inst : SeminormedAddCommGroup E]
  [inst_1 : SeminormedAddCommGroup F] [inst_2 : NontriviallyNormedField 𝕜] [inst_3 : NontriviallyNormedField 𝕜₂]
  [inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F)
  (x : E), ‖f x‖ ≤ ‖f‖ * ‖x‖

The fundamental property of the operator norm: ‖f x‖ ≤ ‖f‖ * ‖x‖.

Defined in

Mathlib.Analysis.Normed.Operator.Basic

Cited by

65 results in Mathlib

Foundations

Depth 164 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroupSeminormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceRingHomIsometric