Theorems · Theorem · functional analysis
ContinuousLinearMap.orthogonal_ker
∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
[inst_2 : NormedAddCommGroup F] [inst_3 : InnerProductSpace 𝕜 E] [inst_4 : InnerProductSpace 𝕜 F]
[inst_5 : CompleteSpace E] [inst_6 : CompleteSpace F] (T : E →L[𝕜] F),
(↑T).kerᗮ = (↑(ContinuousLinearMap.adjoint T)).range.topologicalClosure- Cited by
- 3 results in Mathlib
- Foundations
- Depth 193 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
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
- Submodulestatement and proof · cited by 7,192
- ContinuousLinearMapstatement and proof · cited by 5,352
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- CompleteSpacestatement and proof · cited by 2,532
- le_antisymmproof · cited by 2,068
- Inner.innerproof · cited by 1,089
- LinearMap.rangestatement and proof · cited by 893
- LinearMap.kerstatement and proof · cited by 848
Cited by3
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.orthogonal_rangeproof · cited by 3
- LinearMap.orthogonal_kerproof · cited by 2
- ContinuousLinearMap.ker_le_ker_iff_range_le_rangeproof · cited by 0