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    $\begingroup$ Do you want that limit to just hold pointwise, or do you want something stronger? If you just want it pointwise, the Riesz representation theorem has nothing to do with this--you can find such a net for any $\Phi$ (because you can find a $\phi$ that works for any finite set of $\psi$s). On the other hand, if you restrict to sequences, $\Phi$ does have to be bounded if $V$ is complete, but this is not obvious (it follows from the Banach-Steinhaus theorem). $\endgroup$ Commented Aug 24, 2015 at 15:18
  • $\begingroup$ @EricWofsey: The limit is a limit of numbers, hence there is no weaker or stronger version. How to construct the net? $\endgroup$ Commented Aug 24, 2015 at 16:08
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    $\begingroup$ I think Eric is asking whether you wanted some kind of uniformity with respect to $\psi$ in that limit. (It can be interpreted as a limit of functions of $\psi$ rather than just a limit of numbers.) $\endgroup$ Commented Aug 24, 2015 at 22:56
  • $\begingroup$ @QiaochuYuan: I see. Although asking for more seems interesting, the pointwise construction is enough for me. $\endgroup$ Commented Aug 25, 2015 at 9:14