1. The problem statement, all variables and given/known data Show that the sup norm on R2 is not derived from an inner product on R2. Hint: suppose <x,y> is an inner product on R2 (not the dot product) and has the property that |x|=<x,y>0.5. Compute <x±y, x±y> and apply to the case x=e1, y=e2. 2. Relevant equations |x|=<x,y>0.5 I've noticed that the notation can vary for the sup norm - in this case |x| is the sup norm. 3. The attempt at a solution My understanding of the sup norm is fragile; in fact, my understanding of linear algebra in general is full of gaps, which is why I'm trying to work through questions in the first chapter of Munkres' Analysis on Manifolds. It looks as though this is set up to be a proof by contradiction, so assuming the hint is true I decided to expand and get to the point where the hint would be useful: <x+y, x+y> = <x, x> + <y, y> + 2<x, y> Using the hint, <x, y> = |x|2, so: <x+y, x+y> = ||x||2 + ||y||2 + 2|x|2 ||x+y||2 = ||x||2 + ||y||2 + 2|x|2 Using the fact that x=e1 and y=e2, this gives 2 = 4. Since the expansion is correct under the definition of inner product, the assumption given in the hint must be incorrect. So, this means that the sup norm does not follow from an inner product, or at least this particular inner product; case closed! Or is it? My concern is that it doesn't seem to address the full question, since at a glance it only appears to address the case when x=e1 and y=e2, so hypothetically, using this argument, could it be possible another x, y satisfies? Or is the fact that we're using a basis for R2 enough that the argument is generalized by its usage?