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