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fa.useful evaluation – Bochner integral in a Fréchet area Answer

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fa.useful evaluation – Bochner integral in a Fréchet area

I’ve a Fréchet area $V$ whose topology is (if it helps) induced by a household $mathcal{P}$ of norms – not simply seminorms – and on this area I’ve a Borel likelihood touchstone $nu$. Now, I might love to behold whether or not it’s workable to make sense of the integral

commence{equation}
int_V x , mathrm{d} nu left( x privilege)
aim{equation}

inside $V$. The touchstone is such that I truly can show that the integral exists as a Bochner integral in some Banach area completions of $V$ with respect to a few of the norms in $mathcal{P}$. But I’m distant from with the ability to show this for all of those norms.

Is there maybe a distinct route to outline Bochner integrals in Fréchet areas? I solely know the habitual Banach area setting and it appears to not breathe sufficient on this illustration.

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