Say that I have a set of $n$ points $N$, and am interested in metrics $d:N\times N \rightarrow \mathbb{R}$ over $N$. Let $M$ denote the set of all metrics over $N$.
Now let me define the distance between two metrics $d_1$ and $d_2$ in $M$ to be: $$\partial(d_1, d_2) = \left|\sum_{i,j \in N}d_1(i,j)-d_2(i,j)\right|$$
It isn't hard to see that $(M, \partial)$ itself forms a metric space.
I am interested in the size of the smallest $\epsilon$-net of $(M, \partial)$. (i.e. the smallest subset $S \subset M$ such that for all $d \in M$ there is some $d' \in S$ such that $\partial(d, d') \leq \epsilon$.)
Are bounds on this quantity known, and/or are there standard techniques for estimating quantities like this?
EDIT: As Suresh points out, there is no finite $\epsilon$-net if we are talking about unbounded metrics. Let us consider normalized metrics $M$ such that for all $d \in M$, and for all $i,j \in N$, $d(i,j) \leq 1$. Of course now for all $d_1,d_2 \in M$, $\partial(d_1,d_2) < n^2$.