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So here is my question: If $X$ is a Banach space that is not reflexive, then $X^*$ is also not reflexive.

I have looked at the answer to the following: A Banach space is reflexive if and only if its dual is reflexive

The issue with the two answers on that post, is that the first has access to the Banach-Alaoglu theorem, which we have not covered yet. The issue with the second is that we do not have the first theorem to use.

The only theorem my text has in relation to reflexive Banach spaces is:

Suppose that $X$ is a reflexive Banach space. Given $\phi\in X^*$ there exists a unit vector $x_0\in X$ such that $|\phi(x_0)|=\|\phi\|$ (that is to say, bounded linear functionals on reflexive Banach spaces attain their norm).

Would anyone be able to help me out with this? I am not sure how this theorem would be used to show that if $X$ is a reflexive Banach space, that the dual of $X$ is reflexive... Thanks.

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user3784030
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1 Answers1

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Suppose $X^*$ is reflexive and that $i_X:X\rightarrow X^{**}$ is not onto. By Hahn-Banach there is a non-zero $\lambda\in X^{***}$ that vanishes on $X$. Since $X^*$ is reflexive $\lambda=i_{X^*}(\ell)$ for some $\ell\in X^*$. Now, note that for each $x\in X,\ 0=\lambda(i_X(x))=\ell(x)$ so $\ell=0$, this contradicts that $\lambda $ is non-zero.

azarel
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