Theorems · Theorem · functional analysis
closedBall_subset_range_iff_surjective
∀ {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]
[SemilinearMapClass 𝓕 σ₁₂ E F] [RingHomSurjective σ₁₂] {f : 𝓕} (x : F) {r : ℝ},
0 < r → (Metric.closedBall x r ⊆ Set.range ⇑f ↔ Function.Surjective ⇑f)- Defined in
- Mathlib.Analysis.Normed.Operator.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 157 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- NormedSpacestatement and proof · cited by 12,499
- RingHomstatement and proof · cited by 10,189
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Set.rangestatement and proof · cited by 4,705
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- FunLikestatement and proof · cited by 2,560
- Metric.closedBallstatement and proof · cited by 704
- RingHomSurjectivestatement and proof · cited by 220
- subset_transproof · cited by 63
Cited by1
Results whose statement or proof uses this declaration.
- sphere_subset_range_iff_surjectiveproof · cited by 0