# Preduals of Banach spaces and in particular of $\text{BMO}(\mathbf R^d)$

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?

-
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: books.google.se/… - 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