How to prove a Banach normed vector space is NOT a Hilbert space?

Question

We know that the Banach space $\big(\Bbb R^n,\|\cdot\|_2\big)$ is a Hilbert space with inner product $\langle x,y\rangle := \sum_{k=1}^n x_ky_k$. However, how to prove that $\big(\Bbb R^n,\|\cdot\|_3\big)$ is NOT a Hilbert space where $\|v\|_3 := \left(\sum_{k=1}^n|v_k|^3\right)^{1/3}$?

More generally is there a characterization of Hilbert spaces other than the existence of an inner product? And supposing $(V,\|\cdot\|_V)$ satisfies this characterization, how can we build an associated inner product?

Answer 1

It's possible to prove that norm comes from inner product if only if Parallelogram law holds, that means:

$$2\|x\|^2+2\|y\|^2=\|x+y\|^2+\|x-y\|^2$$

For all $x,y \in \mathbb{R}^n$.

For example, let $x=(1,0,0,0,\ldots)$, $y=(0,1,0,0,\ldots)$ , then:

$$\|x\|^2=\|y\|^2=1$$

$$\|x+y\|^2=(2)^{\frac{2}{3}}$$

$$\|x-y\|^2=(2)^{\frac{2}{3}}$$

So Parallelogram law doesn't hold.

Answer 2

The completeness properties associated with Banach spaces and Hilbert spaces are not very relevant: a norm on a real vector space is called euclidean if it is induced by an inner product (so a Banach space is euclidean iff it is a Hilbert space). The $p$-norms on $\mathbb{R}^n$ for $p$ other than $2$ can be seen not to be euclidean in lots of ways, e.g., using the parallelogram law as in agha's answer.

To answer your "more generally" questions: there has been a huge cottage industry producing characterisations of euclidean normed spaces. The book by Dan Amir referred to in my minor contribution to this subject gives a huge list. If you know that a normed space is euclidean, then the inner product is given by the formula:

$$ \langle v, w \rangle = \frac{1}{2}(\|v + w\|^2 - \|v\|^2 - \|w\|^2)) $$