# Are there any “other” ways to show a normed space is NOT an inner product space?

If the norm on a real or complex normed vector space is defined by $\|x\| = \sqrt{\langle x,x\rangle}$ where $\langle\cdot,\cdot\rangle$ is an inner product, then the norm satisfies the parallelogram law $$\|x+y\|^2 + \|x-y\|^2 = 2\|x\|^2 + 2\|y\|^2.$$ That's easy to show, and it's a bit more work, but not a lot more, to show that "only if " also holds. (That is done by showing how to define the inner product as a function of the norm, and then showing that that inner product gives you back the same norm.)

If I wanted to prove that the normed space $L^1$ is not an inner product space, I would find a counterexample to the parallelogram law in that space.

So my question is: Is there some reasonable way to write such a proof other than that or things that are in some sense trivially equivalent to showing that the parallelogram law fails?

So, for example, Aronszajn's criterion for a norm to be Euclidean is that the length of two sides and one diagonal of a parallelogram should determine the length of the other diagonal. This is easy to refute in the taxi-cab norm on $\Bbb{R}^2$ without doing the calculations involved in refuting the parallelogram law.