Understanding transitions between vectors and functions

Question

Lately I was reading this fantastic article about how functions can be understood as vectors:

http://www.eng.fsu.edu/~dommelen/quantum/style_a/funcvec.html

I have a couple of questions considering this post:

1) Does a similar reasoning work the other way around, that is, can any vector be understood as a function?

2) It was also mentioned at the end of the article:

(It should be noted that to make the tran­si­tion to in­fi­nite di­men­sions math­e­mat­i­cally mean­ing­ful, you need to im­pose some smooth­ness con­straints on the func­tion. Typ­i­cally, it is re­quired that the func­tion is con­tin­u­ous, or at least in­te­grable in some sense. These de­tails are not im­por­tant for this book.)

Could you explain in a little more detail, how this transition can be done?

3) Can any function be understood as a vector (e.g. a discontinuous one)?

Answer 1

I would like to start this answer by doing a question:

What do we mean when we speak of a vector?

In school, when they first talk to us about vectors, they usually introduce them as arrows that live in the plane, pointing somewhere. In fact, there are two different, very important characteristics of that vector: It's module, and its direction. However, there is a very natural question that someone could ask: Is that vector the only kind of vector? For instance, it is early knowledge that there are different kind of numbers...

The key point to answering that question, is giving a precise definition of what a vector actually is. Doing this, implies having some mathematical knowledge to be able to understand what does giving a precise definition even means. So let's not get into that, and let's try to explain this anyways.

Long story short, a vector is a mathematical object that belongs to a vector space. Ok, so what the %^&# is a vector space? Well, it doesn't matter for now. Take it as something, the same way a number is something, that satisfies certain rules. For example, you can add two vectors and obtain a new vector. However, you can't add a vector with a simple number. You can also multiply a vector by a number, obtaining a longer vector if that number is greater than 1, and a shorter one in the other case.

So the main idea, is that a vector is actually defined by a couple of properties it satisfies, not as being an arrow. So if you say, for instance, a vector is a mathematical object that is color blue, it's shy and has Coronavirus, then anything blue, shy with Coronavirus will be a vector.

A beautiful example of vectors, are functions. There are certain sets of function that may satisfy the properties that define what a vector is. Thus, you can treat them as vectors! It sounds absolutely trivial, but this last sentence is quite profound.

However, you can't include all type of functions, because maybe, there are certain functions you may include that do not satisfy the defining properties of a vector, and thus, they can't be considered as vectors.