CONVERGENT SEQUENCES, CAUCHY SEQUENCES, COMPLETENESS

If one takes quantum mechanics, when they first encounter the wavefunction which is a complex-valued function, they learn that the arena in which quantum mechanics is a Hilbert space. If one goes further in order to understand what a Hilbert space is they find that it is a complete inner product space. While many physicists take advantage of this fact, they do not really interest themselves with what this means in a rigorous mathematical sense. When I first encountered this, I was unsatisfied with the so-called “definition” of a Hilbert space. So I found that I had to learn more advanced mathematics; more specifically, real analysis. To that end, the purpose of this post is to understand what the term “complete” means. To remedy any confusion of what an inner product space is, an inner product space is a vector space $V$ that equipped with an inner product $\langle u, v \rangle$ .

In order to understand what completeness is, we require a couple of definitions:

Definition. Let $\{p_{n}\}_{n=1}^{\infty}$ where $n\in \mathbb{N}$ be a sequence of points in the metric space $(E,d)$ . A point $p\in \mathbb{E}$ is called a limit of the sequence of points if for any $\epsilon>0$ , there exists $N\in \mathbb{N}$ such that if $n>N$ , $d(p,p_{n})< \epsilon$ . If such a limit exists, then we say that the sequence of points $\{p_{n}\}_{n=1}^{\infty}$ converges to the point $p\in E$ .

What this says intuitively, is that in the sequence of points above there exists a term for which $n=N$ which corresponds to the point $p_{N}$ in the metric space $E$ , beyond which any later terms in the sequence will be contained in what we call an open ball which is defined to be the set given by $B_{p}(\epsilon)= \{q\in E|d(q,p)<\epsilon\}$ . We can regard the term $p_{N}$ as a “boundary point”.

Definition. A sequence of points $\{p_{n}\}_{n=1}^{\infty}$ in a metric space $(E,d)$ is said to be a Cauchy sequence, if for any $\epsilon>0$ , there exists $N\in \mathbb{N}$ such that whenever $n,m>N, d(p_{n},p_{m})< \epsilon$ .

The intuitive idea behind this concept is that suppose we take two terms in the sequence of points $\{p_{n}\}_{n=1}^{\infty}$ , we say that it is a Cauchy sequence if whenever these two chosen terms are “beyond the boundary” the distance between these two terms are within $\epsilon$ of each other in the metric space $(E,d)$ .

One important result that I am not going to prove is the following:

Theorem. If $\{p_{n}\}_{n=1}^{\infty}$ is a convergent sequence of points in the metric space $(E,d)$ , then such a sequence is Cauchy.

An important note: the converse of this theorem is not necessarily true. If the converse is indeed true, we get the following definition:

Definition. A metric space $(E,d)$ is said to be complete if every Cauchy sequence of points in the metric space $(E,d)$ converges to a point $p\in E$ .

An example of this is that $\mathbb{R}$ with the metric $d(p,q)=|p-q|$ is a complete metric space. Intuitively, what this means is that given a Cauchy sequence that converges in $\mathbb{R}$ to a real number. In other words, any possible Cauchy sequence will converge to some real number $p$ .

The next post will discuss compactness in the context of metric spaces, covers, and open covers.

Clear Skies!

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