Theorems · Theorem · functional analysis

sphere_subset_range_iff_surjective

∀ {𝕜 : Type u_1} {E : Type u_4} [inst : SeminormedAddCommGroup E] [inst_1 : NontriviallyNormedField 𝕜]
  [inst_2 : NormedSpace 𝕜 E] {F' : Type u_9} {𝓕' : Type u_10} [inst_3 : NormedAddCommGroup F']
  [inst_4 : NormedSpace ℝ F'] [Nontrivial F'] {τ : 𝕜 →+* ℝ} [inst_6 : FunLike 𝓕' E F'] [SemilinearMapClass 𝓕' τ E F']
  [RingHomSurjective τ] {f : 𝓕'} {x : F'} {r : ℝ}, 0 < r → (Metric.sphere x r ⊆ Set.range ⇑f ↔ Function.Surjective ⇑f)

Defined in

Mathlib.Analysis.Normed.Operator.Basic

Cited by

0 results in Mathlib

Foundations

Depth 164 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroupNontriviallyNormedFieldNormedSpaceNormedAddCommGroupNormedSpaceNontrivialFunLikeSemilinearMapClassRingHomSurjective