What are good examples of Banach spaces which are and aren't superreflexive? Whenever properties of Banach spaces like superreflexivity, uniform convexity etc. are discussed, my impression is that almost always the only examples used are $l^p$, $L_{p}$ and $c_{0}$ (at least in places I looked at), but there must be many more interesting Banach spaces to look at  in particular, are there natural (i.e. encountered "in nature", not artificially constructed as counterexamples) examples of reflexive spaces which are not superreflexive?
1 Answer
Probably the most natural examples of reflexive spaces that are not superreflexive are the spaces $(\bigoplus_{n=1}^\infty\ell_q^n)_{\ell_p}$, where $q\in\{1,\infty\}$ and $1<p<\infty$. They are reflexive as the $\ell_p$direct sum of a sequence of finite dimensional spaces, but not superreflexive since they contain isometric copies of $\ell_q^n$ for all $n$.
Tsirelson's space is also an example of a reflexive space that is not superreflexive, but obviously it is not quite so natural since it was invented to be a Banach space having no subspace isomorphic to $\ell_p$ or $c_0$. Figiel and Johnson later constructed a superreflexive space having no subspace isomorphic to $\ell_p$ or $c_0$.

$\begingroup$ I would like to mention that the first example given by Philip appears "in nature" as the following statement in [S. J. Dilworth et al., Constr. Approx. 34 (2011), no. 2, 281–296] shows: "We prove that the Banach space $(\oplus^\infty_{n=1}\ell^n_p)_{\ell_q}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever $1\le p\le\infty$ and $1<q<\infty$". $\endgroup$ Feb 28, 2016 at 5:01