The inner product axioms are the following:
##\text{(a)} \ \langle x+z,y \rangle = \langle x,y \rangle + \langle z,y \rangle##
##\text{(b)} \ \langle cx,y \rangle = c\langle x,y \rangle##
##\text{(c)} \ \overline{\langle x,y \rangle} = \langle y,x \rangle##
##\text{(d)} \ \langle x,x \rangle > 0...