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 (https://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.

Thanks

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.