1. ## Norms help

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')

2. Originally Posted by Josh146
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')
They just wrote it as the inner product with itself (assuming the norm is induced by the inner product) and expanded by bilinearity.

3. Originally Posted by Drexel28
They just wrote it as the inner product with itself (assuming the norm is induced by the inner product) and expanded by bilinearity.
Thank you!