How to make a Hilbert space from scratch? | Part 1/2

Let's create our mathematical world from the beginning while trying to make things consistent (this way of thinking is not based on the history of math, but it is a good practice to think more abstractly).

Now imagine any arbitrary set X. This set includes some “things”, such as oranges, apples, bananas, or, for instance, let's say, real numbers, R. In this post, we assume that the reader is familiar with the definitions of natural numbers and real numbers as sets of numbers. Now, we define a map that takes two elements from this set and assigns a value, d, to them in order to measure the distance between them. There is no limit to what can be used as this distance function. It is crucial, however, that it does not break some rules defined for creating the consistency we need in mathematics.

We call the mapping d(x,y): X * X → [0, +inf) is a metric if, for every x and y that belong to X, we have:

1. d(x,y) = 0 <--> x = y
2. d(x,y) = d(y,x) [symmetry property]
3. d(x,y) =< d(x,z) + d(z,y) [triangle inequality property]

Many metrics fulfill these conditions for each set. For example, we can consider Euclidean distance as a metric for n-dimensional real vectors. Furthermore, the discrete metric, for example, returns 0 when the inputs are identical and 1 otherwise.

Now, the pair of the set X and a metric d, (X,d), creates a new object that we call metric space. We can easily assume that we can have sequences (a string of elements) in X. The sequence can be considered a mapping from the natural numbers, N, to X. For example, (2,5,7,12,19,...) and (1, ½, ⅓, ¼, …) are two sequences in the real number set. These sequences may get closer and closer to a point (regarding a metric) or not. We say that the sequence is convergent to b, as after some points (n >= N), the distance of sequence elements from a certain point (b) can be less than any arbitrary amount. For instance, the first sequence in the example is divergent, and the second one is convergent to 0.

Let's now define another property of some of these sequences. Informally, we can say a sequence is Cauchy if, after some points, the distance between elements is less than any arbitrary value. As you might expect, Cauchy sequences and convergent sequences have some similarities. Actually, when X = R, these are equivalent (as we put it in the axioms of the reals), but not in general. We know that every convergent sequence is Cauchy, but not vice versa. For example, consider X = (0,4] with d(x,y) =|x-y| as the metric. Thus, the harmonic sequence, (1, ½, ⅓, ¼, …), is a Cauchy sequence in this metric space but is not convergent. Since it approaches 0, which is not an element of X.

Hence, there is a subclass of metric spaces called complete metric spaces. A metric space is complete when all the Cauchy sequences are convergent. So I think this is a good point to stop this post and continue the discussion in the next blog post. Please do not hesitate to tell me your opinion or comments about this message, and don’t forget to share the message with your friends if you think it could be useful for them.