Let A,B,C be the standard orthornormal bases of Z^r, Z^s and Z^n respectively and let a,b,c be elements of A,B,C.

Let f be a normed bilinear map, f:Z^r x Z^s -> Z^n. Here's a section of my notes:

Can someone explain the final equality? The previous equality is due to the fact that f is bilinear but I'm not sure how the norm of the sum is evaluated.

('and this equals' comes from the fact that f is a normed map. Ignore 'these two new functions')