These notes introduce vector norms and discuss their properties.
The idea with a vector norm is a to measure the size of a vector with reference to the vector. Such a norm also provides us with a distance measure between vectors, which let us answer questions such as: is close to ?
Let , and . A function that satisfies the following three properties is called a vector norm:
In such cases, we write
The most common vector norm is, by far, the Euclidean norm, also called the 2 norm: Usually, this is the norm that people refer to when they aren't specific about another choice.
A more general norm is the -norm (): This becomes the Euclidean norm when , hence, when it is also called the 2-norm. There are two other common choices:
The above norms are commonly used. A less common norm is: th largest magnitude entries in . } Note that for , this is the -norm defined above, just the largest magnittude entry, and for , this is the -norm.
Does it matter which norm you use? The following theorem helps us understand that something that is small in one norm cannot be arbitrarily large in another.
THEOREM (Equivalence of Norms) Let and be any two vector norms on . Then there exist fixed constants and such that which hold for any .
Example
Vector convergence The importance of this theorem is given by the following definition. Let be a sequence of vectors. We say that converges to if Because of the equivalcence of norms, we can show this result for any vector norm. For the forthcoming problems with PageRank, showing such results with the 1-norm will be especially nice.
A small digression
Many people call these norms the and norms, or even the , and norms. In my view, these are misnomers, although, they aren't incorrect. Usually the and norms apply to sequences and functions, respectively:
(over an appropriately measureable space.) So I prefer 1-norm, 2-norm, and -norm to the "L" versions.