Let $X$ be a Banach space. I think that some time ago I read somewhere that, in general, the space $\ell\_2(X)$ of all sequences $(x\_n)$ in $X$ with $\sum\_{n=1}^\infty \|x\_n\|^2<\infty$ is not isomorphic to the space $L\_2([0,1],X)$ of square integrable $X$-valued functions on $[0,1]$. 


How can I find an example of an space $X$ with $\ell\_2(X)\not\simeq L\_2([0,1],X)$? 



Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$
You can take any Banach space $X$ for which the weak$^\ast$-dentability index $Dz(X)$ is strictly larger than the Szlenk index $Sz(X)$ (note that we have $Dz(X)\geq Sz(X)$ in general). The reasons for this are that:


1. $Sz(\ell\_2(Y))=Sz(Y)$ for any Banach space $Y$ (see Theorem 2.12 of Brooker, *Direct sums and the Szlenk index*, J. Funct. Anal. **260** (2011) 2222–2246);
2. $Dz(Y)\leq Sz(L\_2([0,1],Y))$ for every Banach space $Y$ (see Lemma 1 of G. Lancien *A survey on the Szlenk index and some of its applications*, Rev. R. Acad. Cien. Serie A. Mat. **100 (1-2)** (2006) 209–235); and,
3. the Szlenk index is invariant under isomorphisms.


Indeed, for any such space we have $Sz(\ell\_2(X))=Sz(X)<Dz(X)\leq Sz(L\_2([0,1],X))$. (As an aside, an earlier paper of Gilles Lancien also compares ordinal indices for a Banach space $Y$ and $L\_2([0,1],Y)$; see Lemma 2.6 of Lancien, *On uniformly convex and uniformly Kadec-Klee renormings*, Serdica Math. J. **21** (1995) 1-18.)


Examples of spaces $X$ for which $Dz(X)>Sz(X)$ include non-superreflexive Banach spaces $X$ with $Sz(X)=\omega$; this is because a Banach space is superreflexive if and only if $Dz(X)\leq\omega$ - see Proposition 2.1 of the aforementioned paper *On uniformly convex and uniformly Kadec-Klee renormings* by Gilles Lancien, noting also Theorem 3.7 of the same paper which points out - amongst other things - the fact that a Banach space $Y$ is superreflexive if and only if $Sz(L\_p([0,1],Y))\leq \omega$ for $1<p<\infty$.


Examples of non-superreflexive spaces $X$ with $Sz(X)=\omega$ include:


1. $X= (\oplus\_{n=1}^\infty\ell\_1^n)\_p$ or $X= (\oplus\_{n=1}^\infty\ell\_\infty^n)\_p$ for $1<p<\infty$, since the Szlenk index of the $\ell\_p$-direct sum of a set of finite dimensional spaces has Szlenk index equal to $\omega$.
2. $c\_0$ and other isomorphic preduals of $\ell\_1$ with Szlenk index equal to $\omega$, such as the Bourgain-Delbaen examples from Bourgain and Delbaen, *A class of special $\mathcal{L}\_\infty$-spaces*, Acta Mathematica
**145** (1980), 155-176. Dale Alspach computed the Szlenk indices of the somewhat reflexive Bourgain-Delbaen $\mathcal{L}\_\infty$ spaces, showing that their Szlenk indices are equal to $\omega$ in *The dual of the Bourgain-Delbaen space*, Israel J. Math. **117** (2000) 239-259.
3. The original Tsirelson space $T$ (from Tsirelson's paper) has summable Szlenk index by Proposition 6.7 of Knaust, Odell and Schlumprecht, *On Asymptotic Structure, the Szlenk Index and UKK Properties in Banach Spaces*, Positivity **3** (1999), 173–199, hence $Sz(T)=\omega$.
4. The quasi-reflexive James space $J$ was shown to have Szlenk index $\omega$ by Gilles Lancien in *Indices de Szlenk et espaces de James*, Publications Mathématiques de l’Université Paris 6, Séminaire d’initiation à l’Analyse, 1989/90, n 28.


Using similar ideas one can see that you can also take $X=C([0,\alpha])$, where $\alpha$ is an ordinal less than $\omega^{\omega^\omega}$ and $[0,\alpha]$ is equipped with its (compact Hausdorff) order topology. Indeed, it is known that $Sz(C([0,\alpha]))= \omega^{n+1}$, where $n$ is the (unique finite) ordinal satisfying $\omega^{\omega^n}\leq \alpha<\omega^{\omega^{n+1}}$, but for the same $n$ we have $Dz(C([0,\alpha]))= \omega^{1+n+1}$. The computation of the Szlenk index of $C([0,\alpha])$ is due to Christian Samuel (*Indice de Szlenk des $C(K)$*, Séminaire de Géométrie des Espaces de Banach, Vols.I–II, Publications Mathématiques de l’Université Paris VII, Paris, 1983, pp. 81–91), and the computation of the weak$^\ast$-dentability index is due to P. Hájek, G. Lancien, and A. Procházka (*Weak$^\ast$ dentability index of spaces $C ([0, α])$*, J. Math. Anal. Appl. **353** (2009) 239–243.)


After this, one can take $c\_0(X)$ and $\ell\_q(X)$ for $1<q<\infty$ for the various examples $X$ described above (and various other direct sums) to obtain further examples.


