Manuscript X-C-ST@1 · X-C-ST.tex · lemma · b679954e

Lemma: Formal scalar–density duality

Formal duality and the two defects · lines 1274–1294 · proof included or linked

X-C-ST@1#lem:duality, X-C-ST@1#eq:duality

For compact formal tests $a$ and smooth formal scalars $u$ ,

⟨ F^{♯} a, F^{*} u ⟩ = ⟨ a, u ⟩ (mod κ^{3}) .

Proof. Integration by parts gives $D_{ξ}^{T} = - X_{ξ}$ . Thus

(F^{♯})^{T} = 1 - κ X_{ξ} + κ^{2} (- X_{ζ} + \frac{1}{2} X_{ξ}^{2}),

the inverse of the scalar pullback through this order. Multiplication by $F^{*}$ proves the identity.

Displayed formulas in this passage

X-C-ST@1#eq:duality · line 1278
$⟨ F^{♯} a, F^{*} u ⟩ = ⟨ a, u ⟩ (mod κ^{3}) .$
unlabeled · line 1287
$(F^{♯})^{T} = 1 - κ X_{ξ} + κ^{2} (- X_{ζ} + \frac{1}{2} X_{ξ}^{2}),$