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.
All the other different brands of towers, many using other brands of bricks, might not conform to the vaguely intuitive ‘physics’ I’ve set up here. However, all of them, including the Gowers Towers, have the following three properties: the zero tower made up with only zero blocks, is the most stable, having an instability of zero; if we double each number on the blocks (making them heavier and thinner), we double the instability; and if we add the numbers on one tower to those on another tower, and build a third tower, then its stability is less than the sum of the stabilities of the two original towers (which, unfortunately, is not what happens when just you slide two towers together, unless we fudge the lengths of each block).
Lowering and emulating
The first property to keep in mind, I’ll call the lowering property, is that if you remove any blocks from a tower (in a sleek and skillful way, of course) the tower will drop down and be at least equally stable as before or more stable. Many other brands of tower that are naturally occurring in the wild also have this lowering property. 2
The second property of the Gowers Towers is the most remarkable, and was part of Tim Gowers’ body of work for which he won his fields medal in 1998. Let us say that, in general, a brand of tower is emulatable if, for every possible tower, we can build an emulation: another tower with some building restriction, which has a similar stability. Two simple examples of building restrictions are: every tower must start with a foundation of two zero blocks; or each block must be repeated twice in a row. By similar stability, we mean that there is some finite constant such that the stability of each emulation is between and times the stability of the original. 3
The Gowers Towers are not emulatable. This is much more surprising if you know just how very emulatable other brands of towers are. To put it another way, the Gowers Towers were a market-leader for tower brands having ‘very little symmetry’. To give you some idea of its innovativeness, the question of whether or not such a thing existed was an open for over 60 years: Gowers’ example solved a problem that Stefan Banach had posed in 1932. 4
Raising the stakes
Now let’s play a game. I’m going to try to convince you that, knowing the above facts, it’s a sucker’s game.
You start by choosing a raising move that consists of inserting zero blocks below some of the blocks. For instance, you might choose to carefully place one zero block below what were the 7th and 6th blocks, and two new zero blocks below the 4th block. This is the opposite action to lowering. 5
Then on my turn, I must choose a tower with the set international standard level of stability (a stability equal to 1), such that when you perform your already-chosen move on my specific tower, it goes above a stability of and collapses. If the tower collapses I win, otherwise you do. Unfortunately, for you I can always pick a tower that will lead to my certain victory by collapsing straight after your move. But, you say, “Ha! Let’s raise by moving the construction to somewhere less windy. Like… the moon! There’s no way you can win now.” Alas, even if we took to be a billion, I know there will be a way for me to win, and my cry of “Jenga!” will echo down the hall (also used as an exclamation when any large pile of stuff tumbles sensationally).
Proof of concept
To convince you, let’s suppose you could win for every possible Gowers tower for some fixed threshold . That means that your move increases the instability of every tower by at most times (it’s actually enough to consider only towers with a stability of one: you can multiply the number on each block by 1/stability to get such a tower of unit stability, and undo this multiplication after you’ve done the other moves). We have
for all possible towers.
Suppose I then did a lowering that reversed your raising move. I’ve now got back to the original tower with the original stability. So we know that
for all possible towers.
Putting both inequalities together we have
This means that using your raising move, we can emulate the tower: the stability of the new raised tower is always similar to that of the original. Here, the building restriction is that there are zeros in the same positions as you inserted in your move. But this contradicts the fabulous property of the Gowers tower! That means I can always choose some tower that will collapse when subjected to your raising.
This result is surely simpler to prove directly than it is to prove the stronger non-emulatability of the Gowers tower, but given that the first property is easy to show, and the second is well-known, it seems a shame to do lots of extra work when this simple argument will suffice. If I had wanted to get extra insight into Gowers’ construction, it may have been useful to show the lack of ‘stable raisings’ from the complicated definition. But one of the ways I hoped to make use of the construction didn’t pan out, so the only result needing it in my thesis was fairly inconsequential. 7
Technical notes about Banach spaces
If you want to find out more about Gowers’ solution to the hyperplane and other nasty Banach sequence spaces, start with this note that Gowers wrote explaining the context. The original paper of Gowers, is “A solution to Banach’s hyperplane problem” (££). However a later paper by Gowers and Maurey puts this Banach space into wider context by building it using the same basic construction technique as the Gowers–Maurey spaces. I’ve created a tiny primer to Gowers’ hyperplane counterexample by extracting material from my thesis, which repeats the material in the post above in a more serious and standard manner.
1. Different brands of towers are different Banach sequence spaces. The blocks here represent the unit vector basis, and the property noted here is 1-unconditionality. This is in stark contrast to the similarly constructed Gowers–Maurey spaces which fail to have any unconditional basic sequences. Don’t confuse these blocks with a block basis!
2. The lowering property means that generalised left shifts with respect to the unit vector basis are contractive. These generalised left shifts are a certain type of spreads. Spreads are the foundation of the various construction of Gowers and Maurey. Any space with a symmetric basis (like or ) or subsymmetric basis has (like Garling’s space) has this property, as do many other spaces with unconditional bases such as Schreier space and Tsirelson space, and the James space.
3. An emulation of a tower is a strict subspace isomorphic to the whole space.
4. Gowers’ result proved something strong than Banach’s original question which was “is every inﬁnite dimensional Banach space isomorphic to its closed hyperplanes (co-dimension one subspaces)?”.
5. Lowering was a general left shift, so raising must be a general right shift. Banach Spaces with symmetric and subsymmetric unit vector bases bases have all possible right shifts bounded; spaces such as Schreier space and Tsirelson space have some, but not all.
6. for all would imply that .
7. I showed that a James-type space based on this Gowers space, instead of , doesn’t have block right shifts: we can’t copy and insert terms rightwards as in the traditional James space. Operators such as are not bounded.