TITLE: Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$
QUESTION [11 upvotes]: 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 $\ell_2(X)\not\simeq L_2([0,1],X)$?

REPLY [4 votes]: Non-reflexive Grothendieck spaces are examples of spaces for which $\ell_2(X)\not\cong L_2(X)$. The same holds for all $p\in (1,\infty)$.

Particular examples of such spaces include $X=\ell_\infty(\Gamma)$ for any index set $\Gamma$ (and all infinite-dimensional injective Banach spaces) as well as $X=\mathscr{B}(\ell_2)$.
Suppose that $X$ is a Grothendieck space which is not reflexive. For every $p\in [1,\infty)$ the space $L_p([0,1], X)$ is not Grothendieck. This is a result of Díaz; see Theorem 2 in:

S. Díaz, Grothendieck's property in $L_p(X)$, Glasgow Math. J., 37, 3 (1995), 379–382.

On the other hand, the Grothendieck property is preserved by $\ell_p$-sums ($p\in (1,\infty)$).