Theorems · Theorem · functional analysis

ContinuousLinearMap.opENorm_le_bound

∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NontriviallyNormedField 𝕜₂] [inst_2 : SeminormedAddCommGroup E] [inst_3 : SeminormedAddCommGroup F]
  [inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [inst_6 : RingHomIsometric σ₁₂]
  (f : E →SL[σ₁₂] F) {M : ENNReal}, (∀ (x : E), ‖f x‖ₑ ≤ M * ‖x‖ₑ) → ‖f‖ₑ ≤ M

If one controls the enorm of every f x, then one controls the enorm of f.

Defined in

Mathlib.Analysis.Normed.Operator.NNNorm

Cited by

3 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldSeminormedAddCommGroupSeminormedAddCommGroupNormedSpaceNormedSpaceRingHomIsometric