[Merged by Bors] - feat(Analysis/InnerProductSpace/Positive): U.starProjection ≤ V.starProjection iff U ≤ V
#27162
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Co-authored-by: Thomas Browning <tb65536@users.noreply.github.com>
j-loreaux
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| theorem IsSymmetricProjection.sub_of_comp_eq_right {p q : E →ₗ[𝕜] E} | ||
| (hp : p.IsSymmetricProjection) (hq : q.IsSymmetricProjection) (hqp : q ∘ₗ p = p) : | ||
| (q - p).IsSymmetricProjection := by | ||
| refine ⟨hp.isIdempotentElem.sub hq.isIdempotentElem (LinearMap.ext fun x => ext_inner_left 𝕜 | ||
| fun y => ?_) hqp, hq.isSymmetric.sub hp.isSymmetric⟩ | ||
| simp_rw [Module.End.mul_apply, ← hp.isSymmetric _, ← hq.isSymmetric _, ← comp_apply, hqp] | ||
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| theorem IsSymmetricProjection.sub_of_comp_eq_left {p q : E →ₗ[𝕜] E} | ||
| (hp : p.IsSymmetricProjection) (hq : q.IsSymmetricProjection) (hpq : p ∘ₗ q = p) : | ||
| (q - p).IsSymmetricProjection := by | ||
| refine hp.sub_of_comp_eq_right hq <| LinearMap.ext fun x => ext_inner_left 𝕜 fun y => ?_ | ||
| simp_rw [comp_apply, ← hq.isSymmetric _, ← hp.isSymmetric _, ← comp_apply, hpq] |
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@j-loreaux, should we just have one version of this instead as well?
IsSymmetricProjection.sub_of_range_le_range: range p ≤ range q implies q - p is a symmetric projection?
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I changed it, let me know if I should change it back
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…Projection` iff `U ≤ V` (#27162) For symmetric projections `p` and `q`, `p ≤ q` iff `range p ≤ range q`, so then: `U.starProjection ≤ V.starProjection` iff `U ≤ V`, which implies that `Submodule.starProjection` is an injective function.
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U.starProjection ≤ V.starProjection iff U ≤ VU.starProjection ≤ V.starProjection iff U ≤ V
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For symmetric projections
pandq,p ≤ qiffrange p ≤ range q, so then:U.starProjection ≤ V.starProjectioniffU ≤ V, which implies thatSubmodule.starProjectionis an injective function.vin star projection range iff‖T v‖ = ‖v‖#27619