Theorems · Theorem · functional analysis

ContinuousLinearMap.opNorm_le_iff

∀ {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}
  {M : ℝ}, 0 ≤ M → (‖f‖ ≤ M ↔ ∀ (x : E), ‖f x‖ ≤ M * ‖x‖)

Defined in

Mathlib.Analysis.Normed.Operator.Basic

Cited by

4 results in Mathlib

Foundations

Depth 167 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroupSeminormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceRingHomIsometric