Normalization is the process of taking an element of a normed linear space giving it a
specific norm. Usually one chooses to make the norm equal to one, so, without loss of generality, that
will be the case discussed here. In an inner product space, this can be restated as choosing a vector
which is parallel to the first one but has a length of 1. In principle this process is
quite simple. A fundamental property of the normed linear space is that given an element
v and scalar s and denoting the norm of an element
w as ||w||:
||s w|| = |s| ||w||
Thus, to normalize v to a new element n, we simply
have
n = v/||v||
The constant 1/||v|| is often referred to as the normalization constant.
Ok, so the next question is why would you want to do that? Well, when working
with inner product spaces, normalizing the vector gives you a new vector that basically
indicates only the direction, so that the inner product of another vector
w with n gives the projection of w
onto n (the component of w in the n
direction).
Normalization can also be important in infinite dimensional vector spaces. The set of integrable functions on an interval (the classical Banach space L1) is one example. In this case, the norm of an element f is defined as
||f|| = ∫ |f(x)| dx over the interval of the space. If |f(x)| is supposed to represent the probability distribution for a random variable then normalization becomes important, because the integral of |f(x)| over the whole interval must be equal to one, signifying that one of the possible outcomes must occur. In the case of quantum mechanics, the probability distribution for the outcome of a measurement comes from another function, the wavefunction of the quantum state. According to the Born statistical interpretation, the probability distribution P(x) = |ψ(x)|2. So, the requirement that the probabilities add up to one then implies that ψ must be normalized but now using the norm ||f||2 = ∫ |ψ(x)|2 dx. This means that wavefunctions are elements of a different normed space, the classical Banach space L2. Such functions are often said to be square integrable.
In many cases normalizing an element of a normed linear space is fairly easy; however, in the case of the spaces of functions, calculating the norm of an element can become quite tricky, especially if the interval is infinite in extent.
I should also note that sometimes if you're dealing with some normed quantity that evolves in time, that time evolution will cause it's norm to change. After you get the evolved quantity, you might normalize it again. When they do this, people sometimes say they have "renormalized" the quantity. In quantum field theory there is also a procedure called renormalization, but it is entirely different, having to do with certain infinite contributions to the energy of the system that must be removed mathematically in order to get sensible results.