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).In the fourth, verbally I only said: and : if you have any way of describing that in layman’s terms that includes the case , then I’d love to hear about it.
However, even though what we said seemed trivial, now we’ve formalised the concept in a technical mathematical language, we can:
- Prove non-obvious statements about the easy cases.
- Use the same methods to prove results for cases that don’t fit into this (slightly imaginary) motivating framework. For instance, instead of points, we can think about distances between functions like or (functions can be thought of as vectors too).
- Prove statements about all possible norms at the same time (even the ones we haven’t imagined yet).
- Write complicated statements in a precise, clear and communicable way.
Vectors have to live somewhere: we call a suitable collection of vectors that go together a vector space. In the definitions above both and belong to the same vector space, call it X.
The number is a scalar, often a real number in but sometimes a complex number in , or perhaps in another field depending on the vector space you are building on.
The norm is a norm on the vector space . It is a function from to .
I’ve given too many assumptions: I didn’t need to say that or that if . But I think it’s easier to accept it as excess baggage in the definition, especially when teaching.