Hilbert Space: Treating Functions as Vectors

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 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.

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).

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.

I could be misunderstanding it, though.

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.

Consider a function on R as an |R|-dimensional vector...

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"

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.

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.

The point is that we need some way to deal with objects that are inherently infinite-dimensional.

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.

You can look at use cases for an index, and see how well they hold up.

Asking where the smallest greater number (next number) is no longer makes sense.

Taking two numbers and asking whether one is greater than the other still makes sense. (and hence also whether they are equal)

Taking two numbers and asking how far separated from each other still makes sense.

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.

> It fixates on one particular basis and it results in a vector space with few applications and it can not explain many of the most important function vector spaces, which are of course the L^p spaces.

Except just about all relevant applications that exist in computer science and physics where fixating on a representation is the standard.

Haha, this works if you already know what a vector space is. But I think pedagogy needs to provide motivational examples. I'll quote one section of a text by Poincaré (translated by an LLM since most here do not speak French).

> We are in a geometry class. The teacher dictates: “A circle is the locus of points in the plane that are at the same distance from an interior point called the center.” The good student writes this sentence in his notebook; the bad student draws little stick figures in it; but neither one has understood. So the teacher takes the chalk and draws a circle on the board. “Ah!” think the students, “why didn’t he say right away: a circle is a round shape — we would have understood.”

> No doubt, it is the teacher who is right. The students’ definition would have been worthless, since it could not have served for any demonstration, and above all because it would not have given them the salutary habit of analyzing their conceptions. But they should be shown that they do not understand what they think they understand, and led to recognize the crudeness of their primitive notion, to desire on their own that it be refined and improved.

The learning comes from making the mistake and being corrected, not from being taught the definition, I think.

Anyway, it's from Science and Method, Book 2 https://fr.wikisource.org/wiki/Science_et_m%C3%A9thode/Livre...

There's more to the section that talks about the subject. I just find this particular paragraph amusingly germane.