x is a vector of
And in this case, we have
the xi's are the coordinates of the vector.
1) Take the absolute value of each of them.
2) Take the power of each.
3) Sum them all.
4) Take the root of that sum.
For example, if p= 1, that is just the sum of the absolute values.
If p= 2, it is just the usual "Euclidean" norm on .
If p= 3 , n= 4, , , , and , the norm is .
If p= 2 or any even power, you don't need the absolute value. But with odd p, odd powers could cancel- and we don't want that. (1, -1) is not the 0 vector so it shouldn't have 0 norm. But if we used p= 3 without the absolute value, we would have . With the absolute value that becomes