Rigged Hilbert Space in Quantum Mechanics and Generalized Notion of Sequence

Question

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.

Answer 1

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.

Answer 2

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?