This post is an attempt to communicate some of the feel of Banach space theory to those who aren’t familiar with it. I once tried to explain my research to a six year old using Jenga blocks, but fortunately only got as far as the triangle inequality. Near the end of my Phd, at my supervisor’s suggestion, I started to explore the complicated Banach space that is Timothy Gowers’ solution to Banach’s hyperplane problem. These experiences inspired the following explanation of one relatively simple observation (that I included as an example in my thesis) through the delightful medium of building blocks.
Our object of study are towers of good old-fashioned building blocks. Each block has a number written on its side, so each tower built from these blocks gives a sequences of numbers . These don’t have to be positive natural numbers, but you won’t lose much by pretending, in this post, that they are. There are innumerably many different brands of towers, but we’ll concentrate on one particular brand: the ‘Gowers Towers’. Let’s say the number written on each block represents how heavy the block is, and is inversely proportional to the length of the block. So we’d represent the sequence with the Gowers Tower pictured.
It’s worth mentioning that the Gowers Towers include every individual tower of finite height that you can build with your unlimited set of Gowers branded building blocks (and lots of infinite height, but you don’t really need to worry about those here).
Let’s pretend we’ve got a measure of the instability of a tower (the norm of the sequence), and whenever we increase the instability beyond a certain threshold, , the tower collapses.
Blocks with higher numbers are heavier, as well as narrower and perhaps inherently more unstable. How the blocks of different weights at different heights affect the stability of the Towers of Gowers is extremely complicated. However, the towers do have some nice, intuitive properties.
In Part I of Getting into Norms, I talked about three different ways of measuring distance (I also considered the accuracy of a series of guesses to be a ‘distance’). All three of these were norms, but there are many ways of measuring distances that aren’t norms.
So to study norms, mathematicians must define them really rigourously, using something known as axioms. These are the basic assumptions and definitions of mathematics. Once we’ve made these assumptions we can prove what has to follow from them.
We can think of norms as a measure of distance from the origin. If you think about it in this way, the following seem quite obvious, and appeal well to our instincts. A norm satisfies the following three axioms.
- Distances are always positive!
- If the distance from your location to the origin is zero, then you must be at the origin. Or alternatively, if two points are separate then the distance between them isn’t zero. Conversely, the distance from any point to itself is zero.
- Taking a detour is always longer than travelling in a straight line. This is the triangle inequality: the sum of the length of any two sides of a triangle is longer than the length of the third.
- Now we come to axiom four. This one is tough to describe in words. Here goes. If you walk a pace forwards and then take another in the same direction, then you will have walked twice the distance of the original pace. Also it doesn’t matter whether you take a pace forwards or backwards: they will give you the same distance.
When mathematicians want to be precise, we use symbols. The distance between points and is written as . The distance from to the origin is . We say that is a norm if whenever we pick vectors and , and a number , then the following axioms hold:
- If then . And visa-versa.
These four conditions should match with our verbal descriptions above. You may recognise them from this blog’s exquisitely hand-drawn logo.
They were pretty trivial intuitions, once we thought of as being the distance of a point from the origin (the origin above is underlined to distinguish it from the normal , though we don’t choose a different notation because the origin behaves a lot like the number zero). Continue reading