Let $E$ be a uniformly convex Banach space and $p \in (1, \infty)$. Is $L_p(X, \mu, E)$ uniformly convex?

Question

Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p := L_p(X, \mu, E)$. If $E = \mathbb R$ then $L_p$ is uniformly convex.

Is $L_p$ uniformly convex if $E$ is uniformly convex?

Any reference for the proof is greatly appreciated.

Answer 1

I remember seeing the proof on this in a peper while ago. I do not remember the title of the paper, but the proof is done in three steps:

• $l^p(E) = \left\{\left(x_n\right)_{n\in\mathbb N}:\, x_n\in E,\; \sum_{n\in \mathbb N} \left\|x_n\right\|^p < \infty\right\}$ is uniformly convex.
• For fixed $X_1, X_2, \ldots, X_n$, $\mu-$measarable sets, $$\left\{\sum_{i=1}^n h_i \mathbf 1_{X_i}: h_i \in E\right\}$$

is uniformly convex. (use step one)

• Finally use the density of simple functions in $L^p(X,\mu, E)$ to prove your statement.