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I have been reading Arno Bohm and Israel Gel'Fand, on Rigged Hilbert Space ($\Phi \subset H \subset \Phi'$), which is an attempt to put the Quantum Mechanics Bra-Ket notation, of Dirac, on rigorous mathematical footing. This makes heavy use of linear topological spaces, which in turn makes heavy use of the notion of sequences and their convergence.

According to A. Bohm ("The Rigged Hilbert Space and Quantum Mechanics", 1978, P. 31) "$\Phi'$ is a space in which the topology cannot be completely described by the description of the passage to the limit of countable sequences ($\Phi'$ does not satisfy the first axiom of countability and is, therefore, a more general topological space...)"

My experience with sequences in the literature is that they always assume an integer index, that is, the sequence $\phi_n$ has n being integer. For instance, we have {$\phi_0$, $\phi_1$, $\phi_2$, $\phi_3$,...,} where the value of $\phi_n$, for any integer $n$, may be an integer, real, complex, etc. Indeed, Wikipedia (http://en.wikipedia.org/wiki/Sequence) defines a sequence to be a "a function whose domain is a convex subset of the set of integers".

My question is does any branch of analysis define sequences $\phi_x$ where $x$ is real and not limited to integers? I was thinking that such a generalization of the notion of sequence might be useful for defining a useful topology for $\Phi'$.

My apologies if this belongs more in math.stackexchange, but my motivation for the question was based on reading physics.

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There are two (almost) equivalent notions of generalized sequences that are useful to study the topology of non-countable spaces: nets and filters.

The nets are a more immediate generalization of sequences, as they use a directed set instead of natural numbers to parametrize the sequence. A directed set is a partially ordered set such that every finite subset has an upper bound. Given a directed set $A$, a net $(x_{\alpha})_{\alpha\in A}$ is a function from $A$ to some set $X$. Most of the usual results involving sequences can be extended to nets. One important difference is the concept of a subnet. One has to be careful with subnets: for example, a sequence is a net ($\mathbb{N}$ is a directed set), but its subnets may not be sequences! In fact the only requirement for $(y_\beta)_{\beta\in B}$ to be a subnet of $(x_{\alpha})_{\alpha\in A}$ is that there exists a function with cofinal image and order preserving $f:B\to A$ such that $y_{\beta}=x_{h(\beta)}$. As a matter of fact there are topological spaces where compactness arguments require the extraction of a subnet even from a given sequence.

Filters are a more fundamental set-theoretical concept, and historically they are more used at least in the french school of general topology (see e.g. Bourbaki's books). I will not discuss them in detail here since the question was explicitly about generalized sequences.

Finally, let me point out that topological vector spaces are uniform spaces, and as such the most important concept to grasp in order to understand their topology are entourages and neighbourhood bases. A good reference that also clearly explains the importance of dual topologies and such is the book Topological Vector Spaces by Bourbaki.

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  • $\begingroup$ @David I extended my answer a little bit, and added a maybe interesting reference. $\endgroup$ – yuggib Jan 12 '17 at 7:52
  • $\begingroup$ @ Yuggib I am wondering if cauchy nets would be useful in Rigged Hilbert Spaces (AKA Gel'Fand Triple), especially regarding the $\Phi'$ (of $\Phi \subset H \subset \Phi'$) which is the conjugate of the Schwartz function space and does not satisfy the first axiom of countability. I have never seen cauchy nets, or net in general, applied to Rigged Hilbert Spaces. $\endgroup$ – David Jan 12 '17 at 19:21
  • $\begingroup$ @David it depends what you want to do... $\Phi'$ can be endowed with many topologies (ultraweak, weak,etc.). The point is whether you need or not the topological properties of the dual. Usually on distributions one does not look at topological aspects. $\endgroup$ – yuggib Jan 12 '17 at 20:53
  • $\begingroup$ According to Gel'Fand, "Generalized Functions", V2, P 57 (paraphrasing): A sequence in $\Phi'$ 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 $\Phi'_0 \subset \Phi'_1 ...\subset \Phi_p... \subset \Phi'$. It seems restrictive to only talk about convergence of sequences where all the elements of the converging sequence have to be in the same 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? $\endgroup$ – David Jan 12 '17 at 21:12
  • $\begingroup$ What is $\Phi'_p $? Anyways it seems that he's just discussing a sufficient to have convergence. In any case, it may happen that a sequence does not converge but a subnet does converge. The point is that you need nets mostly to study compact sets and the alike. The important thing is always which topology you are considering, and the convergence of what you need to prove. $\endgroup$ – yuggib Jan 12 '17 at 21:26
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A sequence can be otherwise considered to be a function $f : \mathbb N \to S$ such that $a_n = f(n)$.

If you instead want to index over the reals, you have $f : \mathbb R \to S$ such that $a_x = f(x)$.

Does this look like something from topology you already know how to work with?

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  • $\begingroup$ Thanks! I am reading Gel'Fand's series on Generalized Functions and he talks about Rigged Hilbert Space and the complexities of defining a topology for the conjugate of the Schwartz Space ("$\Phi'$")because it does not satisfy the first axiom of countability. I was wondering if indexing over the reals in defining sequences and their convergence might be a more natural approach to defining a topology for $\Phi'$ $\endgroup$ – David Jan 12 '17 at 7:28

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