Multiplication is a binary operation. It takes two inputs and generates one output: ##m\, : \,(p,q) \longmapsto p\cdot q.## A function ##f## is linear if ##f(\alpha x+\beta y)=\alpha f(x) +\beta f(y).## Now, let's look at the first input variable of multiplication: ##m(\alpha x+\beta y, q)=(\alpha x+\beta y)\cdot q = \alpha x\cdot q +\beta y \cdot q =\alpha m(x,q)+\beta m(y,q).## This means ##m(\, . \,,q)## is linear in the first argument. The same is true for the second argument, so ##m(p,\, . \,)## is linear, too. This means that ##m(\, . \,,\, . \,)## is linear in both arguments, i.e. it is bilinear.