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At page 103 of Brezis's Functional Analysis there is

The following table summarizes the main properties of the space $L_p(X, \mu, \mathbb R)$ when $X$ is a measurable subset of $\mathbb{R}^N$ : $$ \begin{array}{|c|c|c|c|} \hline & Reflexive & Separable & Dual space \\ \hline L^p \textit{ with } 1<p<\infty & YES & YES & L^{q} \\ \hline L^1 & NO & YES & L^{\infty} \\ \hline L^{\infty} & NO & NO & \textit{Strictly bigger than } L^1 \\ \hline \end{array} $$

Here $p^{-1}+q^{-1}=1$. I would like to make an analogue for Banach spaces.

Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Fix $p \in [1, \infty)$. Let $L_p := L_p (X, \mu, E)$ and $\|\cdot\|_{L_p}$ be its norm. Here we use the Bochner integral. Then

  1. If $p \in (1, \infty)$ and $E$ reflexive, then so is $L_p$.
  2. If the $\sigma$-algebra $\mathcal A$ is countably generated and $E$ separable, then $L_p$ is separable.
  3. $L_{p}(X, \mu, E)^{*}=L_{q}(X, \mu, E^{*})$ IFF $X^{*}$ has the Radon-Nikodým property with respect to $\mu$.
  4. If $p \in (1, \infty)$, then $L_p$ is uniformly convex IFF $E$ is uniformly convex.

I'm happy to receive your correction and suggestion!

Akira
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  • It is not clear from your posting how the properties of E are reflected in $L_p(\mu;E)$. Is $\Omega$ meant to be $E$? – Mittens Nov 12 '22 at 17:46
  • @OliverDíaz $\Omega$ is a notation used by Brezis. I thought I wrote that "If...$E$ is reflexive, then so...". – Akira Nov 12 '22 at 17:48
  • @OliverDíaz I have re-arranged my thread so that It is less confusing. – Akira Nov 12 '22 at 17:52
  • I would just replace the omega by E. That way it is clear that you want to see what properties remain from the vector valued $L_p(E)$ when E is Euclidean space the case where $E$ is more general Banach space satisfying some extra conditions. – Mittens Nov 12 '22 at 17:53
  • @OliverDíaz Actually, in Brezis's book $L^p(\Omega) := L_p(\Omega, \mu, \mathbb R)$. – Akira Nov 12 '22 at 17:55
  • Which makes your posting more confusing. I think you want to pass from real finite dimensional valued integrals to Banach space valued integrals and want to see which properties on the Banach space may be required to have something that behaves well. – Mittens Nov 12 '22 at 17:58
  • @OliverDíaz I have just edited again. Now I think there is no more confusion. – Akira Nov 12 '22 at 18:00
  • Let us continue this discussion in chat. – Mittens Nov 12 '22 at 18:01
  • @OliverDíaz: please don't use the chat, it separates the discussion from the question and no one will ever look at it. – Martin Argerami Nov 12 '22 at 22:11
  • @MartinArgerami: Usually I try to avoid it, but sometimes Mods force long threads into chat. But I understand... – Mittens Nov 12 '22 at 22:34
  • I know that sometimes mods do, but reality is that it separates the information from the question it came from. Already the system shortens the list of comments on display, so removing the comments when there is no issue with any particular comment seems pointless. – Martin Argerami Nov 12 '22 at 22:41

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