Frechet derivative of bilinear map

Question

Let $f:X \times Y\rightarrow Z $ is a bilinear map, I'm trying to calculate the Frechet derivative using the definition. I've tried the following with $(x,y), (h,k) \in X \times Y $. So $$\|f(x+h,y+k)-f(x,y)-A_{(x,y)}(h,k)\|_Z=\|f(x,k)+f(h,y)+f(h,k)-A_{(x,y)}(h,k)\|_Z $$ Now if we set $A_{(x,y)}(h,k)=f(x,k)+f(h,y)+f(h,k) $ then the expressions above are 0 and so surely A is the frechet derivative, I know this isn't right but why is this the case? ($X, Y, Z $ are normed spaces)

Answer 1

Because the function $A_{(x,y)}$ is not linear. Observe that for $(a,b),(c,d)\in X\times Y$ we have that $$A_{(x,y)}((a,b)+(c,d))=A_{(x,y)}(a+c,b+d)\neq A_{(x,y)}(a,b)+A_{(x,y)}(c,d)$$