Let
V,W,L be
vector spaces over a
field k. A function
f:VxW-->L
is
bilinear if it is linear in each variable. That is
f(av+bv',cw+dw')=acf(v,w)+adf(v,w')+bcf(v',w)+bdf(v',w') (*)
for
a,b,c,d in
k and
v in
V
and
w in
W.
Equivalently, for each v in V and each
w in W the induced functions
W-->L defined by x|-->f(v,x)
and V-->L defined by y|-->f(y,w)
are linear transformations.
More generally, if V,W,L are modules over a commutative ring
k then we can use (*) to define bilinearity in this
more general context.