Theorems · Theorem · functional analysis
ContinuousLinearMap.IsStarNormal.orthogonal_range
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{T : E →L[𝕜] E} [inst_3 : CompleteSpace E], IsStarNormal T → (↑T).rangeᗮ = (↑T).kerThe range of a normal operator is pairwise orthogonal to its kernel.
This is a weaker version of LinearMap.IsSymmetric.orthogonal_range
but with stronger type class assumptions (i.e., CompleteSpace).
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 199 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.
- RingHom.idstatement and proof · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- Submodulestatement · 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 · cited by 893
- LinearMap.kerstatement · cited by 848
- ContinuousLinearMap.toLinearMapstatement · cited by 528
- Submodule.orthogonalstatement · cited by 257
- IsStarNormalstatement and proof · cited by 117
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