From: israel@math.ubc.ca (Robert Israel)
Newsgroups: sci.math
Subject: Re: Simple question on Banach spaces.
Date: 14 May 1998 19:33:42 GMT
In article , rob@amsta.leeds.ac.uk (Rob Clother) writes:
|> Take a Banch space $X$.
|>
|> Does the identity
|>
|> X^{***} = X^*
|>
|> always hold? If so, why? If not, are there any obvious counterexamples?
No. Your identity says X^* is reflexive, but X is reflexive if and only if
X^* is reflexive: see e.g. Rudin, Functional Analysis, chap. 4 exercise 1(f).
So any non-reflexive Banach space is a counterexample.
Robert Israel israel@math.ubc.ca
Department of Mathematics (604) 822-3629
University of British Columbia fax 822-6074
Vancouver, BC, Canada V6T 1Z2
==============================================================================
From: edgar@math.ohio-state.edu (G. A. Edgar)
Newsgroups: sci.math
Subject: Re: Simple question on Banach spaces.
Date: Mon, 18 May 1998 10:03:41 -0400
> edgar@math.ohio-state.edu (G. A. Edgar) wrote:
>
> > However, I think the James space J may be constructed without choice,
> > and in that example J***/J* has dimension 1, so J*** is not equal to J* .
> > This counterexample does not require AC.
In article <2g4syp6h1y.fsf@hera.wku.edu>, Allen Adler
replied:
> What is the definition of the James space?
>
See Lindenstraus & Tzafriri, CLASSICAL BANACH SPACES I, page 25.
This is also an example where J is isometric to J**, but J is
not reflexive.
--
Gerald A. Edgar edgar@math.ohio-state.edu