#metric space

[aromagelily]

If I see one more fucking video talking about "non euclidian' math in explicitly euclidian environments I'm gonna scream

[mathkonspekt]

An Introduction to Functional Analysis on Coursera

Proff. John Cagnol from École Centrale Paris teach this course carefully. Starting with topological and metric spaces he finishes with Sobolev spaces and PDE. 

This the course starts September 12th.

[mathkonspekt]

Lecture 2. Metric and Normed spaces

This is my lecture notes on An Introduction to Functional Analysis.

←Lecture 1

[whatilearninmath]

my analysis teacher's games

part 3 of discrete math (toc)

Sometimes, professors have to teach classes that they aren't so interested in. So I guess, to make the time pass faster, they throw a little bit of their own games into the mix. My analysis professor introduced us to p-adic numbers. Let's look at them and what the beans they're used for, shall we?

Step 1: cut a whole in the box. Express a - b as the following: a - b = (p^x)(m/n) where m, n, p, and x are all integers, m and n both not divisible by p.

Step 2: then, we say d(a, b) = p^(-x).

I'll give you a second to absorb that. Weird, huh? How do we think about this metric? Well, let's start with thinking about distances from zero. Say we pick p = 5. Then we can see that 5, 10, 15, 55 all are the same distance from 0: namely, 1/5. Similarly, 25, 100, and 400 are equidistant from zero, at 1/25. Going the other way, we see that 1/125, 1/250, etc, are all 125 away from 0.

Let's confirm that what we've specified is indeed a metric:

d(a, b) = p^(-x) > 0 if a ≠ b, since p is positive. When a = b, then we get d(a, b) = 0 since we can factor out infinitely many ps from 0. So we're fine.

d(a, b) = d(b, a): yes. We just get b - a = (p^x)(-m/n), which yields the same distance.

d(a, c) ≤ d(a, b) + d(b, c): okay. Let's do some setup. First, we let a - b = (p^x)(m/n). Similarly, we'll let b - c = (p^y)(r/s). On the right-hand side, therefore, we get d(a, b) + d(b, c) = p^(-x) + p^(-y). Now moving onto the left-hand side: a - c = a - b + b - c = (p^x)(m/n) + (p^y)(r/s). WLOG, we'll assume x ≤ y. Then, a - c = (p^x)[(m/n) + (p^(y-x))(r/s)]. Now the sum in the brackets may factor out some more ps. We don't know. Either way, we'll end up with p to some exponent that is greater than x. Once we take the metric, therefore, we'll get something that is smaller than or equal to p^(-x). Therefore, the left hand side is less than or equal to the right hand side.

Now...you may be wondering (as I was when our professor introduced it) what the p-adic metric may be useful for. Well...having already rambled enough about metric spaces, you should be able to predict that we can now impose a new kind of topology onto Q: namely, using the p-adic metric! Q with the p-adic metric is sadly not a complete space (as we attempted to prove in our homework), so the completion of Q under the p-adic metric yields the p-adic numbers. These numbers are all the rationals plus a few irrationals that result from the completion--these vary depending on which prime we pick for p.

Great, now we have Qp, the completion of the rationals under some p-adic metric. What does that do for us now? Around here our professor stopped. But I did a brief search and here is one application I get: enter diophantine equations, which are basically equations for which only integer solutions are allowed. People who like to solve diophantine equations like to use p-adic numbers. If you show that an equation has no solutions in the 2-adic numbers, for example, then clearly the solution must be irrational--and thus, not an integer. Surprisingly, this kind of argument crops up a lot when people show that certain diophantine equations are unsolvable. Using any p-adic numbers will do the trick, so you also get flexibility depending on your equation.

Diophantine equations seem pretty inane to me, but I can imagine practical applications: for example, how to section off computer memory, which must be done in integer partitions. That's why we leave this tedious stuff to the computer engineers.

I've catalogued this post under discrete math, because diophantine equations seems pretty discrete to me, and that's the application I'm giving for p-adic numbers. If you would like to hurt your brain, you can try and think about what an open (or closed) set looks like under the p-adic metric. My classmates and I tried very hard and unsuccessfully on multiple occasions. and that reason is why this post is most decidedly NOT under real analysis.

(also thanks, p-adic numbers, for being the questions I couldn't answer on the exams D:)

[whatilearninmath]

closed, open, and clopen: yes, it's possible.

part 1 of the series on metric space topology. (toc)

We did a whole bunch of stuff recently. We're going to put it all together today with some METRIC SPACE TOPOLOGY WOO!!! (Metric spaces are just sets/spaces with an endowed metric, ie a way of measuring distances. In the real numbers, the distance between a and b is |b - a|; I wrote a post touching on alternative metrics if you guys want to check that stuff out...part 3 of the series.)

The last part of the puzzle is what we call a limit point. A limit point is basically a really popular point. It's got lots of friends. In fact, it's never alone and it's impossible to separate it from its friends.

First off, we're going to define a neighbourhood. The neighbourhood of size r around a point p is just all the points, q, such that d(p, q) < r. In 3D euclidean space, that's a sphere around the point of radius r. In 2D, a circle. In 1D, an interval.

SHUTUPMISSROWAN. Okay. A limit point of a space X is any point, p, where all neighbourhoods around p contain other points of X. In other words, you can't find a radius where N(p, r) contains no points of X. Obviously, that also means that all neighbourhoods of p have infinitely many points of X! Note that the limit point does not necessarily have to be contained in the space.

Here's the crutch: the value to which a sequence converges is a limit point, almost by definition. Remember we defined the value of convergence as the number, a, to which the sequence an gets closer and closer. No matter what e > 0, we could find N where |an - a| < e, for n > N. Thus, every neighbourhood of a has infinitely many points from X: namely, the terms in the sequence an. There is no e where this neighbourhood would be empty.

Likewise, suprema (and infima) are also limit points. If we took a bounded subset of X, say E, and identified its supremum as b, then all neighbourhoods of b will contain elements from E. Recall that if there were a radius where the neighbourhood of b didn't contain an element of E, then b is not the supremum. Likewise for the infimum.

We define a "closed" set as a set that contains all its limit points. Thus, in closed sets, all cauchy sequences will converge and the least upper bound property holds. The opposite direction is also clearly true: if we want all cauchy sequences to converge (and the least upper bound property to hold) for a space, then the space needs to have all its limit points--so it must be closed.

In essence, R is the closure of Q. Wow, that was an easier way to construct R! Not nearly as cool as our other way, though.

But...if there are closed sets, then there must be open sets! Why yes, there are. And they come with their own characteristic point too: the interior point. An interior point is exactly what it sounds like: it's a point that is enveloped snugly in the set. Formally, an interior point is a point, p, such that we can find r where N(p, r) is completely contained in the set. In 2D, that means there needs to be a circle around p that is also contained by the set in question. (3D, a sphere; 1D, an interval.)

Open sets are sets where all the points are interior points. Graphically, that means that open sets cannot include their boundaries. Boundary points cannot be interior points by definition.

Simple examples: [0, 1] is closed. (0, 1) is open. The language is consistent with the language used to describe intervals because intervals are just subsets of R. Another result about open and closed intervals is that the complement of an open set is closed; and vice versa. 

Now, the burning question...can sets be both open and closed? Well--yes. But usually the examples are pretty strange. The simplest is the empty set. It has no points, so it has no limit points; so it contains its limit points vacuously. Similarly, it has no points, so all its points are interior points. (Actually, the empty set has some pretty kickass properties--unexpected, for something so, well, empty.) And since the complement of a closed set is open and the complement of an open set is closed, then the complement of the empty set must also be clopen. Thus, any metric space is clopen with respect to itself.

Clopen clopen clopen...sounds like the sound a horse would make.

[isomorphismes]

More Quasimetrics

A list of quasimetrics from real life:

one-way street

temperature preferences

ratchet

running distance around the track

magnitude

downside risk in a financial portfolio

Facebook EdgeRank

progress

time

I love you, but you like me back. Ouch. ("He loves me in his own way", however, is not on a 1-D scale against "like" -- so a functional to a quasimetric would pervert the meaning.)

[isomorphismes]

More Quasimetrics

A list of quasimetrics from real life:

one-way street

temperature preferences

ratchet

running distance around the track

magnitude

downside risk in a financial portfolio

Facebook EdgeRank

progress

DNA sequences

time

arXiv papers on quasimetrics (to read):

The Geodesic Problem in Quasimetric Spaces

Quasi-metric and metric spaces

Quasi-metric spaces with measure

Quasi-Metric Relativity