Theorems · Theorem · functional analysis
ContinuousLinearMap.orthogonal_range
∀ {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).rangeᗮ = (↑(ContinuousLinearMap.adjoint T)).ker- Cited by
- 3 results in Mathlib
- Foundations
- Depth 194 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
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
- LinearMap.rangestatement and proof · cited by 893
- LinearMap.kerstatement and proof · cited by 848
- LinearIsometryEquivstatement · cited by 748
- starRingEndstatement · cited by 671
Cited by3
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.norm_eq_iInf_range_iff_adjoint_apply_eq_zeroproof · cited by 1
- LinearMap.orthogonal_rangeproof · cited by 0
- ContinuousLinearMap.IsStarNormal.orthogonal_rangeproof · cited by 0