Today’s A To Z term is another free choice. So I’m picking a term from the world of … mathematics. There are a lot of norms out there. Many are specialized to particular roles, such as looking at complex-valued numbers, or vectors, or matrices, or polynomials.
Still they share things in common, and that’s what this essay is for. And I’ve brushed up against the topic before.
The norm, also, has nothing particular to do with “normal”. “Normal” is an adjective which attaches to every noun in mathematics. This is security for me as while these A-To-Z sequences may run out of X and Y and W letters, I will never be short of N’s.
A “norm” is the size of whatever kind of thing you’re working with. You can see where this is something we look for. It’s easy to look at two things and wonder which is the smaller.
There are many norms, even for one set of things. Some seem compelling. For the real numbers, we usually let the absolute value do this work. By “usually” I mean “I don’t remember ever seeing a different one except from someone introducing the idea of other norms”. For a complex-valued number, it’s usually the square root of the sum of the square of the real part and the square of the imaginary coefficient. For a vector, it’s usually the square root of the vector dot-product with itself. (Dot product is this binary operation that is like multiplication, if you squint, for vectors.) Again, these, the “usually” means “always except when someone’s trying to make a point”.
Which is why we have the convention that there is a “the norm” for a kind of operation. The norm dignified as “the” is usually the one that looks as much as possible like the way we find distances between two points on a plane. I assume this is because we bring our intuition about everyday geometry to mathematical structures. You know how it is. Given an infinity of possible choices we take the one that seems least difficult.
Every sort of thing which can have a norm, that I can think of, is a vector space. This might be my failing imagination. It may also be that it’s quite easy to have a vector space. A vector space is a collection of things with some rules. Those rules are about adding the things inside the vector space, and multiplying the things in the vector space by scalars. These rules are not difficult requirements to meet. So a lot of mathematical structures are vector spaces, and the things inside them are vectors.
A norm is a function that has these vectors as its domain, and the non-negative real numbers as its range. And there are three rules that it has to meet. So. Give me a vector ‘u’ and a vector ‘v’. I’ll also need a scalar, ‘a. Then the function f is a norm when:
- . This is a famous rule, called the triangle inequality. You know how in a triangle, the sum of the lengths of any two legs is greater than the length of the third leg? That’s the rule at work here.
- . This doesn’t have so snappy a name. Sorry. It’s something about being homogeneous, at least.
- If then u has to be the additive identity, the vector that works like zero does.
Norms take on many shapes. They depend on the kind of thing we measure, and what we find interesting about those things. Some are familiar. Look at a Euclidean space, with Cartesian coordinates, so that we might write something like (3, 4) to describe a point. The “the norm” for this, called the Euclidean norm or the L2 norm, is the square root of the sum of the squares of the coordinates. So, 5. But there are other norms. The L1 norm is the sum of the absolute values of all the coefficients; here, 7. The L∞ norm is the largest single absolute value of any coefficient; here, 4.
A polynomial, meanwhile? Write it out as . Take the absolute value of each of these terms. Then … you have choices. You could take those absolute values and add them up. That’s the L1 polynomial norm. Take those absolute values and square them, then add those squares, and take the square root of that sum. That’s the L2 norm. Take the largest absolute value of any of these coefficients. That’s the L∞ norm.
These don’t look so different, even though points in space and polynomials seem to be different things. We designed the tool. We want it not to be weirder than it has to be. When we try to put a norm on a new kind of thing, we look for a norm that resembles the old kind of thing. For example, when we want to define the norm of a matrix, we’ll typically rely on a norm we’ve already found for a vector. At least to set up the matrix norm; in practice, we might do a calculation that doesn’t explicitly use a vector’s norm, but gives us the same answer.
If we have a norm for some vector space, then we have an idea of distance. We can say how far apart two vectors are. It’s the norm of the difference between the vectors. This is called defining a metric on the vector space. A metric is that sense of how far apart two things are. What keeps a norm and a metric from being the same thing is that it’s possible to come up with a metric that doesn’t match any sensible norm.
It’s always possible to use a norm to define a metric, though. Doing that promotes our normed vector space to the dignified status of a “metric space”. Many of the spaces we find interesting enough to work in are such metric spaces. It’s hard to think of doing without some idea of size.
I’ve made it through one more week without missing deadline! This and all the other Fall 2019 A To Z posts should be at this link. I remain open for subjects for the letters Q through T, and would appreciate nominations at this link. Thank you for reading and I’ll fill out the rest of this week with reminders of old A-to-Z essays.