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In general the predual of a Banach space is not unique. If there are multiple ones must they be isomorphic?

More specifically is $H^1(\mathbf R^d)$ the only predual of $\text{BMO}(\mathbf R^d)$ or are there any others? If there are others, are they isomorphic?

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If you don't get an answer, I'd be very tempted to ask this over at MathOverflow-- I suspect it's quite a hard problem. – Matthew Daws Oct 20 '11 at 9:44
No, not at all in general. Some examples that show how bad the situation can be were given in this MO thread. I don't know about $\operatorname{BMO}$ off-hand. – t.b. Oct 20 '11 at 9:47
@Matt: part of it was answered in the thread I just linked to. Do you mean the BMO-stuff should be hard? Why? – t.b. Oct 20 '11 at 9:48
Here are some related results:… - the focus is on analytic spaces though. – AD. Oct 20 '11 at 10:34
@t.b. That is a nice result as well. I have found in Triebel's book about function spaces that there are a lot of preduals to $H^1$. I think they are called Triebel-Lizorkin spaces, but Triebel doesn't name them like that ($F$-spaces). However, that does not give a complete characterisation (if it exists). – Jonas Teuwen Oct 20 '11 at 12:36

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