Theorems · Theorem · functional analysis
ClosedSubmodule.iInf_orthogonal
∀ {𝕜 : Type u_4} {E : Type u_5} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{ι : Type u_7} (K : ι → ClosedSubmodule 𝕜 E), ⨅ i, (K i)ᗮ = (iSup K)ᗮThe inf of an indexed family of orthogonal subspaces equals the subspace orthogonal to the sup.
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- Foundations
- Depth 182 from the axioms · uses propext, Classical.choice, Quot.sound
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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- NormedAddCommGroupstatement and proof · cited by 15,752
- InnerProductSpacestatement and proof · cited by 3,523
- RCLikestatement and proof · cited by 2,829
- iSupstatement · cited by 2,415
- iInfstatement · cited by 1,690
- ClosedSubmodulestatement and proof · cited by 123
- GaloisConnection.l_iSupproof · cited by 78
- ClosedSubmodule.orthogonalstatement · cited by 32
- ClosedSubmodule.orthogonal_gcproof · cited by 4
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