The Multi-Index Notation

We begin with the standard multi-index notation in the modern theory of partial differential equations. Let x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn) be points in the Euclidean space Rn. Then the dot product x · y of x and y is defined by

x · y = n∑

xjyj

and the norm |x| of x is given by

|x| =  n∑

1/2 . Let α = (α1, α2, . . . , αn), where the entries α1, α2, . . . , αn are nonnegative

integers. Then we call α a multi-index and we define its length |α| by

|α| = n∑

αj .