Theorems · Theorem · functional analysis
ContinuousLinearMap.isClosed_range_iff_antilipschitz_of_injective
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_5} {F : Type u_6} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] [CompleteSpace E]
[CompleteSpace F] (f : E →L[𝕜] F), Function.Injective ⇑f → (IsClosed (Set.range ⇑f) ↔ ∃ K, AntilipschitzWith K ⇑f)An injective bounded linear operator between Banach spaces has closed range iff it is anti-Lipschitz.
- Defined in
- Mathlib.Analysis.Normed.Operator.Banach
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites14
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
- RingHom.idstatement and proof · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- ContinuousLinearMapstatement and proof · cited by 5,352
- Set.rangestatement and proof · cited by 4,705
- NNRealstatement and proof · cited by 4,310
- CompleteSpacestatement and proof · cited by 2,532
- IsClosedstatement and proof · cited by 1,639
- AntilipschitzWithstatement and proof · cited by 132
- ContinuousLinearMap.uniformContinuousproof · cited by 12
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