$Reflexive Banach Space and Duality$

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$Reflexive Banach Space and Duality$

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.

Functional Analysis

Bradz

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1. If $X$ is reflexive, then $X^{\ast\ast} = X$ by definition, so we have \[ (X^\ast)^{\ast\ast} = (X^{\ast\ast})^\ast = X^\ast \] so $X^\ast$ is reflexive.

2. Since $X^\ast$ is reflexive, by 1. we have that $X^{\ast\ast}$ is reflexive. But $J_X$ is an isomorphism between $X$ and a closed subspace of $X^{\ast\ast}$, and a closed subspace of a reflexive space is reflexive, so $X$ is reflexive.

3. If there exists $u \in L^\infty(E)$ such that $\int_E uv$ does not take a maximum on the closed unit ball in $L^1(E)$, then the unit ball is not compact in the weak topology and so $L^1(E)$ is not reflexive. (I assume this is the result "seen in class", but I'm guessing here).

Erdos

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$Reflexive Banach Space and Duality$

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