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I asked myself the following question when I was student just for curiosity. I asked a bit around (my professor, some researchers that I know), but nobody was able to give me an answer. So maybe it is just that nobody thought enough about that, or maybe it is not a stupid question.

Question: Do there exist two Banach spaces, one separable and one non-separable, having isomorphic dual spaces?

Note: isomorphic in the sense that there exists a linear homeomorphism between the two.

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If you're willing to accept the negation of the axiom of choice, see the answers to this question: mathoverflow.net/questions/5351/… –  Mark Meckes Oct 6 '11 at 18:50
    
I tried to open that link twice and each time my laptop gets mad!! Anyway... ehm, actually I use Zorn's lemma every day.. –  Valerio Capraro Oct 6 '11 at 18:57

3 Answers 3

up vote 19 down vote accepted

The duals of $C[0,1]$ and of $C[0,1]\oplus_\infty c_0(\Bbb{R})$ are isometrically isomorphic.

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Beautifully simple! (At least, if you take the Riesz representation theorem for $C[0,1]^*$ for granted.) –  Mark Meckes Oct 6 '11 at 19:13
    
Professor Johnson, what would be a relatively up to date reference for studying the question of what spaces are duals? I have some survey papers but they are a few decades old. –  Andres Caicedo Oct 6 '11 at 19:14
    
Your question is too broad, Andres. Do you have something more specific in mind? –  Bill Johnson Oct 6 '11 at 19:28
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I agree Mark! When Bill's answer appeared whilst I was writing mine, I couldn't believe I didn't think of that instead. –  Philip Brooker Oct 6 '11 at 19:38
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Andres, although it doesn't tell you which Banach spaces are duals and which are not, if you're wondering whether a given Banach space is a dual space you could look at finding a suitable compact topology on the unit ball, as in Sten Kaijser's paper A note on dual Banach spaces, Math. Scand. 41 (1977), 325--330. –  Philip Brooker Oct 6 '11 at 19:45

The James Tree space $JT$ and $JT \oplus_2 \ell_2(2^{\aleph_0})$ have isomorphic duals.

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Thannk you! Can you please tell me what is the James tree space? –  Valerio Capraro Oct 6 '11 at 19:11
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Valerio, the book Topics in Banach Space Theory by Albiac and Kalton contains lots of the information that you are after, including the definition of the James Tree space. You mentioned in a comment to Bill Johnson's answer that you want to know which spaces are duals; obviously reflexive spaces are, and it is classical (I think due to Civin and Yood?) that quasi-reflexive spaces are duals of quasi-reflexive spaces. Note that a dual space is complemented in its bidual, so for example you can show that $c_0$ is not a dual space by showing that it is not complemented in $\ell_\inty = c_0^{**}$ –  Philip Brooker Oct 6 '11 at 19:32
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That $c_0$ is not complemented in $\ell_\infty$ is shown in the Albiac-Kalton book, but they also give a proof that $c_0$ is not linearly homeomorphic to a subspace of a separable dual space (so therefore isn't a dual itself). They also give a proof that $L_1[0,1]$ does not embed in a separable dual, so $L_1[0,1]$ is not a dual either. Despite this, $L_1[0,1]$ is in fact complemented in its bidual. I strongly recommend the Albiac-Kalton book. –  Philip Brooker Oct 6 '11 at 19:35
    
Sorry Valerio, I now see that it was someone else (Andres Caicedo) who asked about a reference for dual spaces. –  Philip Brooker Oct 6 '11 at 19:39
    
don't worry, it's always important to have the opportunity to learn something. Thank you very much. –  Valerio Capraro Oct 6 '11 at 19:46

To the best of my knowledge, among classical Banach spaces, $c_0,$ C[a,b], $L_1[a,b],$ $l_{\infty}/c_0$ are not dual.

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