Theorems · Theorem · functional analysis

ContinuousLinearMap.exists_approx_preimage_norm_le

∀ {𝕜 : Type u_1} {𝕜' : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NontriviallyNormedField 𝕜'] {σ : 𝕜 →+* 𝕜'}
  {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] {F : Type u_4}
  [inst_4 : NormedAddCommGroup F] [inst_5 : NormedSpace 𝕜' F] (f : E →SL[σ] F) {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ']
  [RingHomIsometric σ] [RingHomIsometric σ'] [CompleteSpace F],
  Function.Surjective ⇑f → ∃ C ≥ 0, ∀ (y : F), ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖

First step of the proof of the Banach open mapping theorem (using completeness of F): by Baire's theorem, there exists a ball in E whose image closure has nonempty interior. Rescaling everything, it follows that any y ∈ F is arbitrarily well approached by images of elements of norm at most C * ‖y‖. For further use, we will only need such an element whose image is within distance ‖y‖/2 of y, to apply an iterative process.

Defined in

Mathlib.Analysis.Normed.Operator.Banach

Cited by

1 results in Mathlib

Foundations

Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceRingHomInvPairRingHomIsometricRingHomIsometricCompleteSpace