Manuscript X-C@3 · X-C-v3.tex · lemma · 2260f5f8

Lemma: Formal duality

C4: completed test, three-term variation, and cancellation › Scalar and density pullbacks · lines 1017–1031 · proof included or linked

X-C@3#lem:duality

Back to manuscript

For compact formal tests $c$ and smooth formal scalars $ψ$ ,

⟨ F^{♯} c, F^{*} ψ ⟩ = ⟨ c, ψ ⟩ .

Proof. $D_{ξ}^{T} = - X_{ξ}$ . Thus $(F^{♯})^{T}$ is the inverse scalar pullback $1 - κ X_{ξ} + κ^{2} (- X_{ζ} + \frac{1}{2} X_{ξ}^{2})$ . Multiplying by $F^{*}$ gives the identity through second order.

Displayed formulas in this passage

unlabeled · line 1020
$⟨ F^{♯} c, F^{*} ψ ⟩ = ⟨ c, ψ ⟩ .$

Nearby in this section

section: Scalar and density pullbacks
definition: Definition: Coordinate and switch defects
section: Scalar and density pullbacks
lemma: Lemma: Coordinate defect