I am having difficulties with understanding some aspects of inner products. For example, ||u||² = <u,u> Where <u,u> denoted the inner product of "u" with itself. My problem here is that we can define any inner product we wish. For example, if I defined, <u,v> = u1v1 + 3(u2v2) Then the above equation for finding the magnitude of "u" doesn't agree with the other formula for finding the magnitude of the vector.. ||u|| = sqrt(u1^2 + u2^2) Why does this happen? There are other equations where this same thing happens. Shouldnt everything agree?