Theorems · Theorem · functional analysis

SemilinearMapClass.bound_of_continuous

∀ {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), Continuous ⇑f → ∃ C, 0 < C ∧ ∀ (x : E), ‖f x‖ ≤ C * ‖x‖

A continuous linear map between seminormed spaces is bounded when the field is nontrivially normed. The continuity ensures boundedness on a ball of some radius ε. The nontriviality of the norm is then used to rescale any element into an element of norm in [ε/C, ε], whose image has a controlled norm. The norm control for the original element follows by rescaling.

Defined in

Mathlib.Analysis.Normed.Operator.Basic

Cited by

2 results in Mathlib

Foundations

Depth 160 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroupSeminormedAddCommGroupNontriviallyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceFunLikeRingHomIsometricSemilinearMapClass