Manuscript XV-B@2 · XV-B.tex · lemma · 4d3f12e8

Lemma: Coefficient extraction

Finite normal jets and complete reconstruction-null tests › Insertion and extraction · lines 756–779 · proof included or linked

XV-B@2#lem:extraction

For a finite family $b_{γ} \in D^{'} (X)$ ,

∣ γ ∣ \leq r \sum E_{γ} b_{γ} = 0 ⟺ b_{γ} = 0 for every ∣ γ ∣ \leq r .

Moreover, testing the sum against $J_{γ} ψ$ extracts $⟨ b_{γ}, ψ ⟩$ .

Proof. Equation (eq:extractjets) and the insertion convention give

⟨ ∣ β ∣ \leq r \sum E_{β} b_{β}, J_{γ} ψ ⟩ = ⟨ b_{γ}, ψ ⟩ .

The forward implication follows for all $ψ$ ; the reverse is immediate.

Displayed formulas in this passage

unlabeled · line 759
$∣ γ ∣ \leq r \sum E_{γ} b_{γ} = 0 ⟺ b_{γ} = 0 for every ∣ γ ∣ \leq r .$
unlabeled · line 770
$⟨ ∣ β ∣ \leq r \sum E_{β} b_{β}, J_{γ} ψ ⟩ = ⟨ b_{γ}, ψ ⟩ .$