Functions are vectors

That the direct conversion from mass to energy follows the same shape isn't really surprising, it sort of has to.

That said, the joule was only explicitly defined as kgm/s^2 in 1946 (or 1935), after Einstein and nuclear physics.

as a side note, one very important technique/idea in mathematics (in general, not just in the area of functional analysis) is describing something in terms of a set of properties that is both as general as possible and as minimal as possible. for instance numbers can be added, subtracted, multiplied and divided, with "obvious" real-world interpretations. mathematicians then asked themselves what properties exactly the numbers had to possess in order for those operations to be defined, and then they proceeded to find other classes of mathematical objects that also had those properties, and suddenly we were able to "add" and "multiply" things that had no obvious physical interpretation for those operations. but since their structure was mapped to the structure of the numbers, those operations could be mechanically defined over them, and you had all sorts of mathematical tools at your disposal.

here a similar thing was done with functions. there had been a lot of work put into studying the operations you could do on "vector spaces", a mathematical structure that generalised the notion of a vector as a collection of numbers. then mathematicians noticed that if you took the minimal collection of properties something needed to have in order to be a vector space, functions satisfied all those properties. and voila - everything that you could prove about vector spaces (and again, it was a whole lot) was suddenly applicable to functions as well.

(why some of this seems a bit tautological is that it also follows the properties of the real numbers, and even non-mathematicians have had a lot of intuition built up about how numbers behave. but it is by no means guaranteed that every mathematical construct will have these same properties.)

Take a vector for normal space, [x, y, z]. We say each component of this vector is one dimension, so this one is 3D, and each of its three components can vary. Two such vectors are different if one or more components differ between them. Treating a function as a vector means treating each distinct possible input to the function as a distinct component.

For example, consider the integer function f(x) = x^2. This can be represented as the vector [..., 16, 9, 4, 0, 4, 9, 16, ...] Where the complete vector has as many components as integers. Since there's infinitely many integers, there's infinite components, so instead of 3D like the three component vector above, this vector is ∞D.

Any single function is representable in this way, so each distinct function has its own unique infinitely long vector.

So each different function is a different "point" in an infinite dimensional vector space.

The formal definition guarantees that whenever a = b, f(a) = f(b). You use it when you need that guarantee.

An ordered pair drawn from ℝ² is in some sense a single value. In another sense, it is two values. Which way you want to think about it depends on what you're going to do with it; if you're thinking about square roots of real numbers, it will be more useful to think of it as two values.

> It seems the parent has just mixed up the domain and codomain, because under that assumption he would be right about both the definition and bijectivity

He still wouldn't be right about bijectivity; you also need the assumption that a function is defined over its entire domain.

The author studied at CMU, where the proudly-paraded slogan for an introductory functional programming classes is "Functions are values", which has an almost cult-like status - appearing on their TA hoodies, laptop stickers, and so on.

Other classes soon caught on, first with the imperative programming class declaring that "Functions are pointers", then the introductory discrete math class's "Functions are tuples", and even "Functions are relations" from the databases class.

So viewed in this lens, passing up the opportunity to title it what it was would have been unthinkable.