Let $(X,\mu)$ be a measure space and let $1<p<\infty$.

Question. Is the space $L^p(X,\ell^p)$, $$ \Vert f\Vert_p=\Big(\int_X\sum_{i=1}^\infty |f_i|^p\, dx\Big)^{1/p}, \qquad f=(f_i)_{i=1}^\infty, $$ uniformly convex?

If the answer if "yes" I would appreciate a reference to the statement/proof.

I know that $L^p(X,\ell^p_M)$, where $\ell^p_M$ is finitely dimensional $\ell^p$ space $$ \Vert f\Vert_p=\Big(\int_X\sum_{i=1}^M |f_i|^p\, dx\Big)^{1/p}, \qquad f=(f_1,\ldots,f_M) $$ is uniformly convex. This result was proved by Clarkson in his original paper where he introduced uniformly convex spaces, see also Uniformly convex Banach spaces. I haven't checked whether the argument applies to the case of values into $\ell^p$.

Let $(X,\mu)$ be a measure space and let $1<p<\infty$.

Question. Is the space $L^p(X,\ell^p)$, $$ \lVert f\rVert_p=\Bigl(\int_X\sum_{i=1}^\infty \lvert f_i\rvert^p\, dx\Bigr)^{1/p}, \qquad f=(f_i)_{i=1}^\infty, $$ uniformly convex?

If the answer if "yes" I would appreciate a reference to the statement/proof.

I know that $L^p(X,\ell^p_M)$, where $\ell^p_M$ is finite dimensional $\ell^p$ space $$ \lVert f\rVert_p=\Bigl(\int_X\sum_{i=1}^M \lvert f_i\rvert^p\, dx\Bigr)^{1/p}, \qquad f=(f_1,\dotsc,f_M) $$ is uniformly convex. This result was proved by Clarkson in Uniformly convex spaces where he introduced the notion; see also Uniformly convex Banach spaces. I haven't checked whether the argument applies to the case of values into $\ell^p$.