Reflexive Banach Space and Duality 

Let $X$ be a Banach space.

1) Show that if $X$ is reflexive, then $X^*$ is reflexive. 
Hint: Consider the adjoint operator of the natural embedding $J_X: X \rightarrow X^{**}$.

2) Conversely, show that if $X^*$ is reflexive, then $X$ is reflexive.
Hint: Remark that the natural embedding provides an isomorphism between $X$ and a closed subspace of $X^{**}$.

3) Let $E \subseteq \mathbb{R}^n$ be a measurable set such that $B(0, r) \subseteq E$ for some $r \gt 0$. Use the results from questions 2 and 3 (attached below) to show that $(L^{\infty}(E), \|\cdot\|_{\infty})$ is not reflexive.


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Erdos Erdos
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