Giu's Journal

Proving That Every Norm Defines a Metric

2012-02-07T19:00:00+01:00

At the end of my previous entry I wrote the following sentence (emphasis mine):

And exactly this is the reason why a norm function ||·|| of a normed vector space (V, ||·||) always induces a metric[...]

This is quite a bold statement to make, and as it is with such a mathematical statement, you need to mathematically proof it.

And so, as an excercise I set myself to work and wrote a proof of the statement on paper (see bottom of this entry), and typed it out afterwards because of possible readability issues.

The statement has been already proven, mind you, and it is not one of the more complicated proofs, but still, it was a nice exercise.

Proof: Every Norm Defines a Metric

Let (V, ||·||) be a normed vector space, with ||·||: V x V → R being the norm and R being the set of real numbers.

By definition of a norm, ||·|| satisfies the following properties:

||a · v|| = |a| · ||v||, a ∈ R, v ∈ V
||u + v|| ≤ ||u|| + ||v|| (triangle inequality)
||v|| ≥ 0 and ||v|| = 0 ⇔ v is the zero vector

We define D_V: V x V → R as D_V(x,y) = ||x - y||.

D_V is a metric if and only if it satisfies the following properties:

D_V(x,y) ≥ 0 and D_V(x,y) = 0 ⇔ x = y (positive definite)
D_V(x,y) = D_V(y,x) (symmetry)
D_V(x,z) ≤ D_V(x,y) + D_V(y,z) (triangle inequality)

In the following steps we will proof that D_V satisfies the properties of a metric.

Proof of 1.): Positive Definite Form

||v|| ≥ 0 ⇒ (Let v = x - y) ||x - y|| ≥ 0

D_V(x,y) = 0 ⇔ (Def. of D_V) ||x-y|| = 0 ⇔ (Def. of norm, 3.)) x-y is the zero vector ⇔ x - y = 0 ⇔ x = y

(Note: While going through my notes again I realized that I can simplify the proof for this step; the proof on paper is a little bit longer)

It follows that D_V is positive definite.

Proof of 2.): Symmetry

D_V(x,y) = ||x-y|| = ||(-1) · (y - x)|| = (Def. of norm, 1.)) |(-1)| · ||y - x|| = ||y - x|| = D_V(y,x)

It follows that D_V is symmetric.

Proof of 3.): Triangle Inequality

D_V(x,z) = ||x - z|| = ||x - y + y - z|| ⇒ (Def. of norm, triangle inequality) ||x - y + y - z|| ≤ ||x - y|| + ||y - z|| ⇒ ||x - z|| ≤ ||x - y|| + ||y - z|| ⇒ D_V(x,y) ≤ D_V(x,y) + D_V(y,z)

It follows that D_V satisfies the triangle inequality.

By having proofed all the three properties of a metric for D_V, it follows that D_V is a metric, and more importantly, that every norm ||·|| defines a metric D_V.

Q.E.D.

Addendum: Proof On Paper

Read Through: 'Die Architektur der Mathematik: Denken in Strukturen' by Pierre Basieux

2012-02-07T00:00:00+01:00

After having read through Gowers's book Mathematics: A Very Short Introduction and gaining some momentum , I decided to read the next book on the subject.

A month or two ago a friend of mine recommended me Basieux's book Die Architektur der Mathematik: Denken in Strukturen, saying it helped him grasping some mathematical concepts we both spent time learning in the past few months better, and so I decided to start reading this book right after having finished reading through Gowers's exemplar almost three weeks ago.

Get the Big Picture

Basieux's book is perfect if you want to learn more about the foundation of math, and more importantly if you want to get a very good overview of it. Just as Gower's book, Basieux's Die Architektur der Mathematik: Denken in Strukturen is a rather slim, but still a quite fascinating exemplar. Since the book's goal is to introduce the reader to the foundation of math and give an overview, it does not go into great detail about the mathematical structures presented, but rather explains the various subjects with just enough details and information, and still manages to teach you a ton of interesting things.

In my humble opinion, sometimes you just really need such an overview that only scratches the surface to fully comprehend a specific subject.

When I bought this book I already had background knowledge about some of the subjects presented, but reading through this book definitely helped me a lot.

I spent the last four months delving into various areas of math at quite a pace, and sadly I missed the one or other connection, because most of the times I did not have a clear picture of the whole thing in front of me.

Just to name you a very, very simple example, a few weeks ago if you would have asked me what the relationship between a metric space and a normed vector space is, I just would have glanced at you with a confused look. In the end it's rather simple, but still, back then I was enough confused with all the other mathematical terms that were thrown at me, I even couldn't understand the rather simple connection between this two terms.

Really, you should have seen my face when I first heard that the norm function induces a metric; it was quite a what the hell does that even mean moment for me.

Now I can tell you why exactly the norm function ||·|| of a normed vector space (V, ||·||) induces a metric, therefore making (V, ||·||) a metric space, too.

To put it simple, let X be a set and let d be a function d: X x X → R (R is the set of real numbers), then the pair (X,d) is called a metric space if d satisfies specific properties, thus making d a metric on X. The norm function ||·|| of a normed vector space (V, ||·||) is a function ||·||: V → R that simply assigns a size to each object of V, and also has to hold some properties. In contrast to the metric d, the norm function ||·|| works on single elements of the set V. Now a function D_V: V x V → R can be defined as D_V(x,y) = ||x -y||, which holds all the necessary properties of a metric thanks to the fact that ||x-y|| = ||x + (-y)||, and thus making D_V a metric on V.

There already exist quite a number of established norms, e.g. the Euclidean or Manhattan norm, but regardless of which norm you are using, be it your own or some established ones, once you define such a norm (implying that it satisfies all the required properties), you can always use it to define a metric, just as shown above.

And exactly this is the reason why a norm function ||·|| of a normed vector space (V, ||·||) always induces a metric, and more importantly, why a normed vector space will always be a metric space, too (the reverse statement, namely that a metric space will always be a normed vector space, too does not always hold).

As I told you before, this is just a simple example, because it was more of a definition issue for me than something else. But still, a lot of things became quite clear for me once I unraveled this knot.

I cannot mention it enough: if you want to get a really good overview of the whole picture, read this book.

Personally, for me reading through this book was a very good and quite eye-opening experience.

But then again, I am just a novice exploring the exciting world of mathematics.

'Die Architektur der Mathematik: Denken in Strukturen' at Amazon.de

Can Math Be Beautiful?

2012-02-07T00:00:00+01:00

You just have to love Euclid's proof of the statement that there are infinitely many prime numbers.

Wikipedia TeX Source Extractor Bug Fix

2012-02-01T00:00:00+01:00

I just pushed an updated version of the Wikipedia TeX Source Extractor userscript to both userscripts.org and GitHub.

The old version had an issue displaying the TeX source of mathematical formulas containing apostrophes ('), e.g. derivations.

If the TeX string contained an apostrophe, the string's part after the apostrophe (including the apostrophe itself) got cropped. For example, if the TeX source behind a derivation looked like f\!\,'(x_0), the Extractor somehow only managed to display f\!\,.

The cause of this issue is my usage of apostrophes for the declaration of strings directly in the JavaScript code.

Because of this conflict with apostrophes, the following snippet of code, in which I inject a textbox containing the TeX source didn't work as expected (shortened version):

var a = $t.attr("alt");
$t.parent().append("<input id='wte_texsource_"+counter+"' type='text' value='"+a+"' />");

To fix the issue, I now first append the textbox without a value, and right after that I inject the TeX source, avoiding any conflicts involving apostrophes:

$t.parent().append("<input id='wte_texsource_"+counter+"' type='text' />");
$("#wte_texsource_"+counter).val($t.attr("alt"));

The updated version now displays any TeX source containing apostrophes correctly:

Read Through: 'Mathematics: A Very Short Introduction' by Timothy Gowers

2012-01-18T00:00:00+01:00

While I was looking for a specific book in one of the math bookshelfs in the Orell Füssli bookstore at Bellevue a few weeks ago, a rather small book managed to catch my attention because of the quite unusual but striking color of its cover (unusual for a math book, that is).

My first thought was simply that somebody deliberately put this book in the wrong shelf, but after picking it up and reading the title I was pleasantly surprised:

I already knew who Timothy Gowers is (he is a prominent British mathematician and recipient of the Fields Medal, the latter sometimes being referred to as the Nobel Prize in Mathematics), since I own another excellent book written by him, namely The Princeton Companion To Mathematics (I haven't read completely through it, though).

The next logical step for me after such a pleasant find was to simply purchase this conspicuously looking book, and I'm really glad I did it.

Small But Mighty

The book Mathematics: A Very Short Introduction, although being an introduction to mathematics (including some advanced topics) is a rather exciting excursion and learning experience in various mathematical topics. It starts with the explanation of how abstraction can be used to build mathematical models of existing systems, and then presents the reader with chapters covering numbers, proofs, limits and infinity, dimension, geometry, estimates and approximates, and ends with some interesting frequently asked questions.

Although it touches the one or other advanced mathematical topic, Mister Gowers definitely managed to do a very good job explaining the various mathematical topics as simple as possible. The various chapters are also spiked with a lot of examples, images and proofs, which help grasp the various presented concepts better.

Personally, I liked the Dimension chapter the most, since it was one of the more revealing chapters for me.

The following is a paraphrased passage from that chapter (I read the German edition of the book):

Imagine we want to draw a simple circle. Drawing a two-dimensional circle is a rather simple task. Add an additional dimension and we are able to draw the equivalent of a circle in the third dimension, namely a sphere. The interesting question now is: what happens if we move one dimension higher and start thinking about drawing a four-dimensional sphere? Is this even possible, and if so, what is this new form called and what does it look like?

This is where the author perfectly brings in the word abstraction. He just describes a circle or a sphere without ever mentioning the dimensions:

A circle or sphere is a set of points with a certain distance from a given point.

Having previously defined a distance function using the same abstraction (no mention of the dimension), we can now simply use the definition of a circle to construct spheres in the fourth or even twenty-seventh dimension.

This chapter was quite the learning experience for me, because a) I never delved into higher-dimensional geometry, and b) as soon as I began reading this chapter I started thinking about the mere impossibility to picture what a four-dimensional sphere may look like, let alone the vast number of things one could do with such a sphere. But right after I read the above-mentioned passage it just clicked. Why trying to do the impossible and imagine what such a figure may look like in higher dimensions? Just abstract its most important properties without ever mentioning the word dimension, and work with the new definition in higher dimensions.

I really enjoyed reading this book. Beside the various interesting topics the book covered, the author did a really good writing job; it is written in quite a captivating style, probably thanks to the fact that it reflects very well Mister Gowers's appreciation for the subject.

This book definitely earns 5 out of 5 stars!

LaTeX: Smoothly Embedding Figures Inside the 'multicols' Environment

2012-01-17T00:00:00+01:00

The presented LaTeX snippet is full of magic, but it will solve your 'Package multicol Warning: Floats and marginpars not allowed inside `multicols' environment!' problem for sure

LaTeX error: 'Package inputenc Error: Unicode char \u8: not set up for use with LaTeX'

2012-01-02T00:00:00+01:00

Solution: Just remove and add all the spaces again

Wikipedia TeX Source Extractor

2011-12-29T00:00:00+01:00

Earlier today I wrote a little GreaseMonkey user script called Wikipedia TeX Source Extractor (WTSE).

If you ever felt the need to copy the TeX source of a formula while reading various mathematical entries on Wikipedia, WTSE will help you achieve that. Once you click on the desired formula, a textbox with the formula's TeX source already selected will appear; you now only have to hit CTRL-C and you're good to go. It's as simple as that.

You can install WTSE from its userscripts.org page, or you can fork it from GitHub.

One Up

2011-12-28T00:00:00+01:00

It happened again: I decided to change my journal's layout, but this time I went further and decided to switch the backend, too. My journal doesn't run on WordPress anymore; instead, it's now served as a bunch of static HTML pages that are generated by the awesome tool that Jekyll is.

If you've never heard of Jekyll, the idea behind it is quite simple: you give Jekyll various types of files (layout files, content files, etc.) written in various markup languages of choice (HTML, Markdown, Textile), and Jekyll will give you a bunch of static HTML pages in return. If you want to learn more about Jekyll, I recommend you to head over to its wiki.

The approach of using Jekyll instead of WordPress brings quite some advantages along that convinced me to do the switch:

Every page is statically served, so the server will be able to a) serve the pages much faster, and b) handle quite a lot of incoming traffic should the need ever arise;
I don't need a database anymore;
I'm using my favourite text editor with my favourite color scheme configured to write this post; I'm feeling all warm and fuzzy right now because of this;
I don't need to login using an unsecure connection to manage my journal, and can use SFTP or even a simple git push (see GitHub User Pages) instead (I'm currently using SFTP), and;
All I need for my new backend to work now are files containing some markup, meaning I can use git to put everything under version control

I've also decided to restart from scratch with this journal, meaning that you won't find most of the entries from the old journal around here anymore. If you're looking for specific entries from the past, don't hesitate to contact me and I'll look what I can do.

My journal is still under construction at this moment, but I guess it's a good start to have something showing up again after the server transfer.