Theorems · Theorem · functional analysis

ContinuousLinearMap.opNorm_le_of_nhds_zero

∀ {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}
  {C : ℝ}, 0 ≤ C → (∀ᶠ (x : E) in nhds 0, ‖f x‖ ≤ C * ‖x‖) → ‖f‖ ≤ C

Defined in

Mathlib.Analysis.Normed.Operator.Basic

Cited by

1 results in Mathlib

Foundations

Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroupSeminormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceRingHomIsometric