# Is $L^p(X,\ell^p)$, $1<p<\infty$, uniformly convex?

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

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 $$\ell^p$$.

• Isn't $L^p(X,\ell^p)$ isometric to $L^p(X\times \mathbb N)$, $X\times \mathbb N$ with the product measure? Apr 21, 2022 at 16:42
• @GiorgioMetafune I think you are correct and I feel like a moron. If you post your comment as n answer, I will accept it. Apr 21, 2022 at 17:44

## 1 Answer

By Proposition 1.2.24 in T. Hytonen, J. Van Neerven, M. Veraar and L. Weis, Analyis in Banach spaces Vol I, Springer, the spaces $$L^p(X; \ell^p)$$ and $$L^p(X\times \mathbb N)$$, $$X \times \mathbb N$$ with the product measure, are isometric (at least when $$\mu$$ is $$\sigma$$-finite). Then the uniform convexity of $$L^p(X; \ell^p)$$ follows from that of $$L^p$$-spaces.