Theorems · Theorem · functional analysis

SemilinearMapClass.bound_of_shell_semi_normed

∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} {𝓕 : Type u_8} [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} [inst_6 : FunLike 𝓕 E F]
  [RingHomIsometric σ₁₂] [SemilinearMapClass 𝓕 σ₁₂ E F] (f : 𝓕) {ε C : ℝ},
  0 < ε →
    ∀ {c : 𝕜}, 1 < ‖c‖ → (∀ (x : E), ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) → ∀ {x : E}, ‖x‖ ≠ 0 → ‖f x‖ ≤ C * ‖x‖

Defined in

Mathlib.Analysis.Normed.Operator.Basic

Cited by

2 results in Mathlib

Foundations

Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroupSeminormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceFunLikeRingHomIsometricSemilinearMapClass