Proposition 6.12 . [01BN] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
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Proposition 6.12 .
For any φ , ψ ∈ ℰ 1 ( X , ω ) \varphi,\psi\in\mathcal{E}^{1}(X,\omega) t ↦ E ω ( ( 1 − t ) φ + t ψ ) t\mapsto E_{\omega}((1-t)\varphi+t\psi) [ 0 , 1 ] [0,1]
(6.9) E ω ′ ( φ ) ⋅ ( ψ − φ ) := d d t | t = 0 + E ω ( ( 1 − t ) φ + t ψ ) = ∫ ( ψ − φ ) MA ( φ ) E^{\prime}_{\omega}(\varphi)\cdot(\psi-\varphi):=\frac{d}{dt}\bigg|_{t=0+}E_{\omega}((1-t)\varphi+t\psi)=\int(\psi-\varphi)\MA(\varphi)
for any φ , ψ ∈ ℰ 1 ( X , ω ) \varphi,\psi\in\mathcal{E}^{1}(X,\omega)