Theorem 7.2 . [01BS]

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Theorem 7.2.

Assume that $\omega$ has the orthogonality property. Then the composition $E_{\omega}\circ P_{\omega}:C^{0}(X)\to\mathbf{R}$ is Gâteaux differentiable, with directional derivatives given by

\frac{d}{dt}\bigg|_{t=0}E_{\omega}\circ P_{\omega}(f+tg)=\int g\,\MA(P_{\omega}(f)).