Do there exist, either in the literature or in folklore, theorems that characterize some particular $\ell^p$ space(s) ($p\not=1,2,\infty$)?

Such a theorem should reveal the particular space(s) as somehow idiosyncratic, in the sense that no obvious modification of the characterization works for general $\ell^p$ spaces.

Thus it would *not* be interesting here to learn, say,
that $\ell^3$ alone has a dual isomorphic to $\ell^{3/2}$;
obviously this just specializes a general fact from the theory of all
the $\ell^p$'s.

thinkthis question might deserve to be made "community wiki", but I am open to persuasion. – Yemon Choi Dec 13 '10 at 7:18`$\ell^{p^2}$`

. – Aaron Meyerowitz Dec 13 '10 at 8:56abstractconnection between $\ell^p$ and $\ell^{p^2}$, you'd have a winner. – David Feldman Dec 13 '10 at 9:02