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I have been reading about Rigged Hilbert Spaces (AKA Gel'Fand Triple). This is characterized by the relationship $\Phi \subset H \subset \Phi'$, where $\Phi$ is Schwartz space, $H$ is Hilbert Space, or $L^2$, and $\Phi'$ is the conjugate to the Schwartz space. $\Phi$ is the space of wave functions (or state vectors) of Quantum Mechanics. $\Phi'$ is the space of the functionals on $\Phi$ and contains the Dirac delta function $\delta (x)$ and plane wave function $e^{ipx}$, which are the eigenfunctions of the position and momentum operators of Quantum Mechanics. Of note, $\Phi'$ does not satisfy the first axiom of countability.

So, it would seem that the concept of Nets (https://en.wikipedia.org/wiki/Net_(mathematics), or something more general than sequences, would be applicable to analyzing $\Phi'$, but I have never seen that in the literature.

Does anyone have any pointers to nets, or something more general than sequences, being used for defining convergence in $\Phi'$, in Rigged Hilbert Space?

Thanks

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    According to Gel'Fand, "Generalized Functions", V2, P 57 (paraphrasing): A sequence in $\Phi'_p$ converges if all the elements of the sequence are in one of the fixed $\Phi'_p$, and they converge in the norm of $\Phi'_p$, where $\Phi'_0 \subset...\subset \Phi'_p \subset...\subset \Phi'$. It seems restrictive to only talk about convergence of sequences where all the elements of the converging sequence have to be in the same $\Phi'_p$ subspace. Maybe if we talk about nets, instead of sequences, all the elements of the net do not have to be in the same $\Phi'_p$ and can still converge? – 2017-01-12
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    The above comment should begin "A sequence in $\Phi'$ converges if all the elements of the sequence...". Given the above comment (and its correction just stated), it seems invalid to talk about the convergence of a sequence of delta functions in $\Phi'$, since all $\Phi'_p$ are normed and so the delta function could not be in any of the $\Phi'_p$, as it is not part of any Hilbert space. But, it seems, to me, like one should be able to talk about the convergence of a sequence of delta functions. Hence, I am confused and wondering if introducing things like nets in $\Phi'$ would help. – 2017-01-14

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