Examples of metric vector spaces but not normed ? Normed but not prehilbertian? What examples (non-trivial) of vector spaces are : * Normed but not prehilbertian ? * Metric but not normed ? In the schoolroom we have seen examples fairly easy. Thank you in advance.

For the second point, just think of a metric not invariant under traslation. For the first, you want that your norm doesn't satisfy the parallelogram law (otherwise you can construct an inner product which induce the same norm). For the first point $\ell^p$ norm are famous examples! for the second :

If we put over the real line the distance: $ d (x, y)= |\log (\frac{x}{y} )|$;

or over $\mathbb{R}^2 $ we can put $ d(x, y)= \|x\|+\|y\|$ with $d(x,x)=0$

We have translation variant metrics