Sequences with Real Indices

Question

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 example, we could have {$\phi_0$, $\phi_1$, $\phi_2$, $\phi_3$,...,} where the value of $\phi_n$, for any integer $n$, may be an integer, real number, complex number, 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 topologies for spaces that do not satisfy the first axiom of countability, such as the conjugate space to the Schwartz space (this space is part of the Rigged Hilbert Space, which is useful in Quantum Mechanics).

Thanks

Answer 1

If $X$ is a set, one defines a sequence $(\phi_n)\subset X$ to be a function $\phi: A\to X$, where $A$ is any subset of $\Bbb N$ such that $\Bbb N \setminus A=\{1,\ldots, t\}$ for some $t\in \Bbb N$. For simplicity, let's just say that a sequence is a function $\Bbb N \to X$. If we want to consider something like $(\phi_{x})$, where $x$ takes real values, it would only make sense that we define such a thing to be a function $\Bbb R \to X$, or for any matter, a function from some element of a special class of subsets of $\Bbb R$ to $X$.

In general, one defines the notion of a "family of things" (see here and here), say for instance a family of elements of a set $X$, to be a function from some set $A$ to $X$, and $A$ is called the indexing set. A "family" is written as $\{x_{a}\}_{a\in A}$ or $(x_a)_{a \in A}$, etc.

If you are merely asking about the notion of indexing by real numbers, then indeed, there are many settings where one invokes indices which are in $\Bbb R$ (or directed sets, or just any set whatsoever). Examples:

• Mollifiers (which are very important in distribution theory).

• See the definition of a locally convex topological vector space, where families of semi-norms are used.

• A net in a topological space is a function from a directed set to $X$. It's usually written as $(x_a)$, where $a$ runs over the directed set.

Answer 2

In topology, there is the notion of nets. A net is a function from a directed set $I$ into your set of consideration. You obtain a sequence, if you choose $I = \mathbb{N}$. The other special case $I = \mathbb{R}$ gives you a "sequence indexed by real numbers".

To cope with arbitrary topological spaces, however, you need index sets which are much larger than $\mathbb{R}$.