> But we can take it even further; what if we allow any real number as an index?<p>How can an uncountably infinite set be used as an index? I was fine with natural numbers (countably infinite) being an index obv, but a real seems a stretch. I get the mathematical definition of a function, but again, this feels like we suddenly lose the plot…
I think getting hung up on words (in this case index) in mathematics is a trap. They are often stretched to their breaking point and you just kind of go with the flow.<p>> When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean — neither more nor less.’<p>> ’The question is,’ said Alice, ‘whether you can make words mean so many different things.’<p>> ’The question is,’ said Humpty Dumpty, ‘which is to be master — that’s all.<p>In mathematics it is the author's privilege.
You can look at use cases for an index, and see how well they hold up.<p>Asking where the smallest greater number (next number) is no longer makes sense.<p>Taking two numbers and asking whether one is greater than the other still makes sense. (and hence also whether they are equal)<p>Taking two numbers and asking how far separated from each other still makes sense.<p>You may already observe some uses for indexes in programming that don't use all of these properties of an index. For example, the index of a hash set "only cares about equality", and "the next index" may be an unfilled address in a hash set.
This Has it's use. The continouus Fourier Transform is is based on that. You are asking what frequencies is this continouus signal made of. Time is normally defined as a real number in that context, but If you have a continouus time you need continouus frequencies to map time space to frequency space. You can think about an Index as a lego Block, that you need to construct Something.
I think that’s why the author put “vector” in quotes. I kind of imagine it as an ephemeral, infinite list where for some real, when we use that real value as an index into our “vector”/function, we get the output value as the item in this infinite, ephemeral list.<p>I think the only thing that matters is that the indices have an ordering (which the reals obviously do) and they aren’t irrational (i.e. they have a finite precision).<p>Imagine you have a real number, say, e.g. 2.4. What stops us from using that as an index into an infinite, infinitely resizable list? 2.4^2 = 5.76. Depending on how fine-grained your application requires you could say 2.41 (=5.8081) is the next index OR 2.5 (=6.25) is the next index we look at or care about.<p>I could be misunderstanding it, though.
A vector is always a vector -- an element of something that satisfies the axioms of a vector space. The author starts with the example of R^n, which is a very particular vector space that is finite-dimensional and comes with a "canonical" basis (0,...,1,...,0). In general, a basis will always exist for any vector space (using the axiom of choice), but there is no need to fix it, unless you do some calculations. The analogy with R^n is the only reason the "indices" are mentioned, and I think this only creates more confusion.<p>> and they aren’t irrational (i.e. they have a finite precision)<p>No, if you want only rational "indices", then your vector space has a countable basis. Interesting vector spaces in analysis are uncountably infinite dimensional. (And for this reason the usual notion of a basis is not very useful in this context.)
> and they aren’t irrational (i.e. they have a finite precision).<p>I'm not sure if I'm misunderstanding what you mean by 'finite precision' but the ordinary meaning of those words would seem to limit it to rational numbers?
We do it all the time. An index is just indicative that there is a mapping (a function), usually from the integers. However we don't use the subscript notation when indexing by a continuum due to the discomfort you describe.<p>The point is that we need some way to deal with objects that are inherently infinite-dimensional.
I'm probably ignorant of how indexes work at a nuts-and-bolts level, but intuitively this seems like a good idea for certain situations. E.g if we want to keep entries in a specific order but don't know ahead of time how many entries will be added between two existing ones. House numbers in areas with a lot of development are an example of the kind of problem this seems ideal to solve, when there's a clear 'order' based on geography but no clear limit on the number of addresses that could be added 'between' existing addresses.
I think you're still describing a countably infinite set: there's a bijection between the natural numbers and the set of houses.<p>One way to think about it is that, even though you're defining an index that permits infinite amounts of subdivision, from any given house there's always a "next house up" in the vector: you can move up one space.<p>In a real-indexed vector, that notion doesn't apply. It's "infinity plus one" all the way down: whatever real value you pick to start with, x, there's no delta small enough to add to it such that there's no number between x and x+d.
> In a real-indexed vector, that notion doesn't apply. It's "infinity plus one" all the way down: whatever real value you pick to start with, x, there's no delta small enough to add to it such that there's no number between x and x+d.<p>Just to clarify, uncountability isn't necessary for this. It's true for the rational numbers too, which are countable.
That’s kind of how I understand it as well.
The only difference of note, I think, is that you can't enumerate the elements. Instead of being able to say "for each element, ..." you'd have to say "for all elements, ...", like the example of vector length defined as an integral over the full number range.
The author is stretching an analogy, it's a price to pay for starting with R^3 as a motivational example.
There is nothing in the general definition of a vector space that requires it's elements to be "indexed"
Consider a function on R as an |R|-dimensional vector...
What do you understand “index” to mean here? To me, a family indexed by some set is mostly just a different notation for, and attitude towards, a function with domain the indexing set.