[mh-cx]

Hmm, he lost me on page nine where "complete" is explained.

> Complete means that every sequence of vectors |a1>, |a2>, ... satisfying lim ...

How are the elements of this sequence related? And why are we only interested in the elements where the index n/m goes to infinity? What does that even mean if the sequence is arbitrary?

That's probably why I also can't make sense of this:

> Loosely speaking, saying that a Hilbert space is complete means that it contains all of its limits.

[stracer]

This is standard mathematical analysis. Infinite sequence of elements may look like it converges to some target element, judging by mutual distances converging to zero. Such sequence is a Cauchy sequence. When the target element actually exists, then the sequence is also convergent. A space where every Cauchy sequence is convergent, is called complete.

Example: if the space is all real numbers except 0, then any sequence of real numbers accumulating around 0 (for howsoever small a distance, there is always infinite number of points closer to 0), the sequence is a Cauchy sequence, but not convergent (because 0 is not present). So that space is not complete (has a hole).

If the space is all real numbers, then the same sequence is also convergent, and the space is complete (no holes).

[azeemba]

It helped me to look up an example of a space that is not "complete".

Turns out, rational numbers are the classic example of a space that is not complete. A sequence of rational numbers can approximate pi but pi itself doesn't exist in the space (since its irrational). So the rational numbers that get closer and closer to pi form a limit to a value that's not in the space.

[Paul-Craft]

That's a great example. To make it concrete, you can take the sequence such that $a_n$ is the $n$-th partial sum of any of the series here that involve only rational numbers: https://en.wikipedia.org/wiki/List_of_formulae_involving_%CF... multiplied by an appropriate constant.

[steppi]

Agreed that this is pretty terse. The sequences they're talking about are called Cauchy sequences [0]. A sequence a_i is Cauchy if for any epsilon, there exists an N such that if m and n are both greater than N, then |a_m - a_n| < epsilon. A classic example, suppose your space is the set of rational numbers, and consider the sequence a0 = 1, a_1 = 1.4, a_2 = 1.41, ... a_n = sqrt(2) up to n digits after the decimal place. You can verify that this is a Cauchy sequence, successive points get arbitrarily close to each other. This means the rational numbers are incomplete, because this Cauchy sequence of rationals doesn't converge to a rational number. It's the real numbers that forms a complete space.

Completeness is required for nice results like the spectral theorem for self-adjoint operators [1] to hold, which is pretty essential for Quantum Mechanics.

[0] https://en.wikipedia.org/wiki/Cauchy_sequence.

[1] https://en.wikipedia.org/wiki/Spectral_theorem

[l33t7332273]

> How are the elements of this sequence related

They are not necessarily related

> And why are we only interested in the elements where the index n/m goes to infinity?

If the sequence is finite, then we don’t really care to discuss the “limit” of the sequence.

> Loosely speaking, saying that a Hilbert space is complete means that it contains all of its limits.

For a set S to not contain all of its limits means you can have an infinite sequence of points (a_n) where each a_n is in S and there is no point a in S so that the sequence is eventually as close to a as you’d like.

More formally, there does not exist a in S so that for any e > 0 we can pick an M so that m > M implies | a_m - a | < e.

You can see how “m > M” gives a formal meaning to “eventually,” and “for any e > 0 … | a_m - a | < e” gives a formal meaning to “as close to a as you’d like.”

[qazxcvbnm]

Imagine a continuous space without holes in them (no holes at all, not even tiny ones).

[btilly]

Putting that into plain English, "If it looks like a sequence is converging, it really is converging to something."

What does it mean to say that the sequence looks like it is converging?

Naively, it means that any two elements far in the sequence are always very close together. We make that intuition more precise by turning it into a challenge-response, "You tell me how close you want the elements to be, I'll tell you how far out to pick your elements." And then we write that mathematically as

    ∀ ϵ > 0                           # You tell me how close
      ∃ N                             # I'll tell you how far
        such that if N < n, m         # so that any 2 elements
           then || a_n - a_m || < ϵ   # will always be that close

Others have given examples about things like the rational numbers. The canonical example of why it matters comes from Fourier series. Joseph Fourier discovered this one. Consider a bunch of functions f(x) over the interval from 0 to 2π. We can create a dot product with f·g equaling the integral from 0 to 2π of f(x)g(x). And the length of a function f is sqrt(f·f)

Joseph Fourier discovered that the following functions are orthonormal (each has length 1, their dot product with each other is 0).

    1/sqrt(2π),
    sin(x)/sqrt(π), sin(2x)/sqrt(π), sin(3x)/sqrt(π), ...
    cos(x)/sqrt(π), cos(2x)/sqrt(π), cos(3x)/sqrt(π), ...

(I hope I have the constants correct...)

Given any function, he could compute a series that we now call the Fourier series that looks like it added up to that function. But there were complications. It added up perfectly for smooth functions with the same value at the edges like x(2π-x). But it also added up except at a few points for things like square waves. This caused a crisis in mathematics because at the time square waves were not considered functions, and nobody had ever realized that adding up infinite series of smooth functions could do such weird things. Their idea of functions was essentially what we would call analytic functions today. Basically things that look like power series. And Mr. Fourier had just shown that the analytic functions are not complete.

In addition to revealing problems in how we understood math, his technique was very, very useful. Because now you just had to figure out the physics of how, say, heat spreads out or vibrating strings vibrate just for those those sin and cos terms, and then you could figure out heat and vibration for ANY function.

Resolving the math problems started many decades of research. The results of which included better definitions of the real numbers, the ϵ-δ definition of a limit (or ϵ-N for a series), new theories of integration, and the idea of Hilbert spaces.