Theorems · Theorem · functional analysis
ContinuousLinearMap.closed_complemented_range_of_isCompl_of_ker_eq_bot
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedSpace 𝕜 E] [CompleteSpace E] {F : Type u_5} [inst_4 : NormedAddCommGroup F] [inst_5 : NormedSpace 𝕜 F]
[CompleteSpace F] (f : E →L[𝕜] F) (G : Submodule 𝕜 F),
IsCompl (↑f).range G → IsClosed ↑G → (↑f).ker = ⊥ → IsClosed ↑(↑f).range- Defined in
- Mathlib.Analysis.Normed.Operator.Banach
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 176 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites22
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- 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
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement and proof · cited by 7,192
- ContinuousLinearMapstatement and proof · cited by 5,352
- Bot.botstatement and proof · cited by 4,720
- CompleteSpacestatement and proof · cited by 2,532
- IsClosedstatement and proof · cited by 1,639
- LinearMap.rangestatement and proof · cited by 893
- LinearMap.kerstatement and proof · cited by 848
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