Not sure if this goes under Math, Physics, or both. But I’m reading this book where there’s a Hilbert space that something something thoughts something something circular aperture. Anyway, I did try googling the math terms, like “complex” space and “imaginary numbers” but was just so much greek greek math. So please explain like I’m five, what’s a Hilbert space? Bonus points if you can explain complex space without equations.
In: Mathematics
A Hilbert space is a special type of space.
What mathematicians call a *space* is just a set of directions. Basically, if you have “one step forward” and “one step to the right”, you have a space! You’ve got directions, you can multiply the directions to move more or less, the entire shebang.
Now for a Hilbert space, you need to add two more conditions:
* It needs to support multiplication (not all spaces do that!)
* If you’ve got a series of numbers that seem to get closer and closer to one number, that final number — called the limit — needs to be inside the space.
For example: if you take the list of number that are 1 divided by something: for 10 it’s 0.1, for 100 it’s 0.01, for a million it’s 0.000001… it gets closer and closer to 0 but **never reaches 0**. So 0 is the limit for the series of numbers “one divided by x”. That means any space that can contain the numbers of the series should also have 0 in it, or it isn’t a Hilbert space.
Whew. Now with all this mathematical mumbo jumbo, what can we do?
Math is a game of definitions and their consequences. Once we define what a Hilbert space is, we can look at how it works and find useful properties.
One of them is right there in the definition: wanna know if a series will have a limit or not? Instead of proving that it does reach a certain number, which is often hard, just prove that it gets closer and closer to it and show that the numbers of your series live in a Hilbert space, which is easy. Boom, job done, you know there’s a solution that exists. Knowing that a solution exists is sometimes more important than the solution itself, especially if you don’t want to waste hours looking for it!
There’s essentially three things required for something to be a Hilbert space. The first is that it’s a space of “vectors”; the second is that it has a “dot product”.
These are understood in a pretty abstract way, so for the vectors part, you just want that you can do similar things to the things in a Hilbert space as you can do to vectors (add them together and multiply them with scalars, with the usual computational laws). And for the dot product part, you just want some product on the space that follows the same computational laws as a dot product.
The third one is the hardest one to explain: the space is “complete”. The vague idea is that there should be nothing missing from the space. As an example, the space of rational numbers (fractions) is not complete: you look at the numbers 3, 3.1, 3.14, 3.141, and so on, and these numbers get closer and closer together. Yet there’s no rational number they approach; the “missing” number they approach is pi, which is irrational. If you do add all the irrational numbers to the rationals and end up with the real numbers, then that space turns out to be complete.
If your Hilbert space is finite-dimensional, then you actually don’t need to worry about the third one: the first two will give you the third in finite dimensions. However, the term is most often used in an infinite-dimensional context, where the last condition has significance.
One pretty commonly used example is the space of all real-valued functions whose square has a finite integral. You can add those functions together (like you can add the function x and the function sin(x) to get x+sin(x)). And for these functions, you actually have a kind of “dot product” which you get by integrating the product of two such functions. (Note; I’m sweeping some slightly advanced details related to the “completeness” -part in this example under the rug).
If you ever need a better understanding of Hilbert Space, start by looking at Euclidean Space, which is just the scientific name for what you use in your everyday life:
If you take a point in your room, you can describe it by how far North it is (that’s a first number), how far Est it is (that’s a second number) and how high it is (that’s a third number).=> Reality is, as far as day-to-day life is concerned, a “3-dimensional Euclidean space”, which simply means that every position can be described by 3 numbers.
Though in maths, we’re rarely interested in points, we’re more frequently interested in “vectors”, which can be understood as “a movement”. To describe a vector, you need the same amount of numbers as for describing a point: in real life, you say how much you move to the North (use negative numbers for South), how much you move to the Est (use negative numbers for West), and how much you move up (use negative numbers for down).
We’re interested in vectors, because now we have 3 kind of mathematical operations we can do with them:
* Addition. Adding two vectors is simply making one movement after the other. If the first vector was (2,0,0) for (North,Est,Altitude) and the second was (1,1,1), then the sum is (3,1,1).
* “Scalar” multiplication, which correspond to amplifying a vector by a given number. So (3,1,1) amplified by a factor 2 gives (6,2,2), and amplified by a factor 0.5 gives (1.5,0.5,0.5) instead.
* “Inner product”, which is the most difficult of the three operations, and at the core of a lot of mathematical theorem. The inner product of two vectors gives a number such that:
* The bigger the vectors are, the bigger the inner product is (in particular, if one of the vector correspond to “no movement at all”, then the inner product is 0).
* The more similar the vectors are, the bigger the inner product is (in particular, the inner product of vectors that goes into perpendicular directions is 0).
Hilbert spaces are simply “what if we needed Euclidean spaces, but with complex numbers instead”. But the core is the same:
* Each point is given by a set of numbers, here complex instead of real. Same for vectors, that can be seen as “movement” in the space.
* You can add vectors to one another.
* You can amplify vectors by multiplying them by any complex number.
* You can make the inner product of two vectors which will give you how “big” the vectors are and how “similar” the vectors are.
And most of the time, you can forget that there are complex numbers, and just think about Hilbert spaces as if they were Euclidean spaces.
And what are complex/imaginary numbers? That’s a complex question! They have a lot of mathematically equivalent definitions, and all their power is that you switch from one definition to another depending on what is more practical, so it’s quite a mess to explain.
From the point of view of physics, a complex number can simply be seen as a real number together with an angle, where every 180° on the angle is equivalent to changing the sign of the real number. It can be seen as “you were in the middle of changing a number into its opposite, but stopped before you finished, so now you are in a weird in-between were the number is no longer positive but not yet negative”. Complex numbers with an angle of 90° are the “imaginary numbers”, that are exactly at the middle point between positive real numbers and negative real numbers.
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