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.
 A: 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
A: Let me work in the context of locally convex topological vector spaces (over $\mathbb R$ or $\mathbb C$); convex spaces for short.
Metrizable convex spaces are characterized by being first countable, or by 
having their topology be generated by a countable set of semi-norms.  
Normable convex spaces are characterized by having a neighborhood $U$ of
the origin such that $\frac{1}{n} U$ forms a n.h. basis of the origin (for $n = 1, 2, 3, \ldots $).
So one can try to find metrizable convex spaces that are not normable by 
looking for spaces whose topology is generated by countably many semi-norms
which are not comparable to one another.

For example, let $V$ denote the space of continuous functions on $\mathbb R^m$ (for some fixed $m$).  Let $B_n$ denote the ball of radius $n$ in $\mathbb R^m$,
and let $q_n$ denote the semi-norm on $V$ defined by $q_n(f) := $ sup of $f$ on
$B_n$.
If we endow $V$ with the topology induced by the semi-norms $q_n$,
then it becomes a meterizable convex space that is not normable.
A: The $\ell_p$ space (sequences of complex numbers that are absolutely $p$-power summable) are all normed spaces but only (pre)Hilbert if $p=2$. 
Any set supports a zillion of possible metric structures (e.g., the discrete metric) and only a handful of those (if the set is a vector space) will be induced by a norm. 
