Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L\_p := L\_p(X, \mu, E)$ be the [Bochner space](https://en.wikipedia.org/wiki/Bochner_space) of all $\mu$-integrable functions $f:X \to E$. Here we use [Bochner integrals](https://en.wikipedia.org/wiki/Bochner_integral). If $E = \mathbb R$ then $L\_p$ is uniformly convex for $p \in (1, \infty)$.



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> Is $L\_p$ uniformly convex if $E$ is uniformly convex for $p \in (1, \infty)$?
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Any reference for the proof is greatly appreciated.




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I saw [this](https://math.stackexchange.com/questions/4513522/let-e-be-a-uniformly-convex-banach-space-and-p-in-1-infty-is-l-px?rq=1) question on MSE which received one answer whose author does not remember the title of the reference paper. So I post it here.



Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?
A reference for this result would be [Some more uniformly convex spaces](https://www.ams.org/journals/bull/1941-47-06/S0002-9904-1941-07499-9/S0002-9904-1941-07499-9.pdf) by Mahlon M. Day, Bull. Amer. Math. Soc. 47(6): 504-507 (June 1941).
(Alternative link at [Project Euclid](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-47/issue-6/Some-more-uniformly-convex-spaces/bams/1183503696.full))


