TITLE: Bilinearity: what does it mean?
QUESTION [22 upvotes]: What does bilinear really mean?  Everytime I heard the word, I think it should be "linear in 2 ways?"
For example, from the definition of inner product (taken from Appendix A of "Wavelets For Computer Graphics" by Stollnitz):
An inner product on a vector space V is any map from $ V \times V $ to $\mathbb{R}$ that is:

Symmetric $ \langle u | v \rangle = \langle v | u \rangle $
Bilinear
$ \langle au + bv | w \rangle = a \langle u | w \rangle + b \langle v | w \rangle $
Positive definite $ \langle u | u \rangle > 0 $ for all $ u \ne 0 $

But how is bilinearity "linear in 2 ways", (if bilinear really does mean $2\times$ linear!)

REPLY [11 votes]: Linearity exists for 1d functions such as $f(r)$, if that function obeys
$f(ar_1+br_2)=af(r_1)+bf(r_2)$
For 2d functions such as $f(r,s)$, the linearity attribute can exist for one dimension, or the other, or both.  If both, then the function is said to be "bilinear"
$f(ar_1+br_2,s)=af(r_1,s)+bf(r_2,s)$
$f(r,as_1+bs_2)=af(r,s_1)+bf(r,s_2)$