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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.2011-10-20
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    No, not at all in general. Some examples that show how bad the situation can be were given [in this MO thread](http://mathoverflow.net/questions/77383/). I don't know about $\operatorname{BMO}$ off-hand.2011-10-20
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    @Matt: part of it was answered in the thread I just linked to. Do you mean the BMO-stuff should be hard? Why?2011-10-20
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    Here are some related results: http://books.google.se/books?id=VWYNmT5E_EUC&pg=PA35&lpg=PA35&dq=predual+bmoa&source=bl&ots=qjoydD8AgI&sig=sGL3BSs5PphKqL-vnEg1BbyLdz8&hl=sv&ei=MfefTqW_GYv64QSvtJT2BA&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDMQ6AEwAw#v=onepage&q=pre&f=false - the focus is on analytic spaces though.2011-10-20
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    It's just a guess, but my impression of this area is that there aren't really any "theorems"-- just lots of particular constructions. So you'd need to do some analysis with the particular space $BMO$ (again, is my guess).2011-10-20
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    @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).2011-10-20
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    Sorry if I'm dense: did you just say $H^1$ and mean $BMO$?2011-10-20
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    Uh, right. Sorry about that.2011-10-20
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    @t.b.: Did you remove your comment? If so, why? It was quite informative. I would probably accept it if you would place your comments as an answer (except when someone can give me a full answer which I doubt will happen).2011-10-20
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    I did because it was silly after all, the spaces I was talking about were reflexive... Give me a little time, I'll research this a bit, but not right now.2011-10-20
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    Just seen this question. In the case $d=1$ there might be something in the book of Wojtaszcyk "Banach spaces for analysts" -- the result I remember is for $H^\infty$, showing that there are non-isomorphic Banach spaces whose duals are isomorphic to $H^\infty$ (although $H^\infty$ has a unique isometric predual). Perhaps there is also something for $BMO=P_+(L^\infty)$?2012-01-06
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    @YemonChoi Thanks! I'll check it out.2012-01-07

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