Theorems · Theorem · functional analysis

ContinuousLinearMap.opNorm_le_bound

∀ {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} (f : E →SL[σ₁₂] F) {M : ℝ},
  0 ≤ M → (∀ (x : E), ‖f x‖ ≤ M * ‖x‖) → ‖f‖ ≤ M

If one controls the norm of every A x, then one controls the norm of A.

Defined in

Mathlib.Analysis.Normed.Operator.Basic

Cited by

59 results in Mathlib

Foundations

Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroupSeminormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpace