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"Normalization" refers to several related processes:

• ("Feature scaling") A set of numbers whose maximum is $$M$$ and minimum is $$m$$ can be converted to the range from $$0$$ to $$1$$ by means of an affine transformation (which amounts to changing their units of measurement) $$x \to (x-m)/(M-m)$$.

• A set of positive numbers $$\{p_i\}$$ representing probabilities or weights can be uniformly rescaled to sum to unity: divide each $$p_i$$ by the sum of all the $$p_i$$.

• Analogously, a distribution (or indeed any non-negative function with a finite nonzero integral) can be normalized to have a unit integral by dividing its values by the integral.

• A vector in a normed linear space is normalized (to unit length) by dividing it by its norm. This is a general procedure encompassing the two preceding operations as special examples.

The range from $$0$$ to $$1$$ can be made from $$0$$ to any desired limit $$\alpha$$ by multiplying a previously unit-normalized value by $$\alpha$$.

Other kinds of operations exist having a similar intent of re-expressing values in a predetermined range. Many of these are nonlinear and tend to be used in specialized settings.