Theorems · Theorem · functional analysis
LinearMap.continuous_of_isClosed_graph
- 1000+ list: Closed graph theorem
∀ {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] (g : E →ₗ[𝕜] F), IsClosed ↑g.graph → Continuous ⇑gThe closed graph theorem : a linear map between two Banach spaces whose graph is closed is continuous.
- Defined in
- Mathlib.Analysis.Normed.Operator.Banach
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites31
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
- LinearMapstatement and proof · cited by 10,215
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement · cited by 7,192
- LinearEquivproof · cited by 3,317
- Continuousstatement · cited by 2,592
- CompleteSpacestatement and proof · cited by 2,532
- IsClosedstatement and proof · cited by 1,639
Cited by2
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.ofIsClosedGraphproof · cited by 2
- LinearMap.continuous_of_seq_closed_graphproof · cited by 1