Re: Weak Topology

From: G. A. Edgar (edgar_at_math.ohio-state.edu)
Date: 03/21/05 

Date: 21 Mar 2005 09:45:00 -0500

In article <d1kkm0$2l3$1@dizzy.math.ohio-state.edu>, Vincent G.
<vincent.g@libertysurf.fr> wrote:

> Hi,
>
> It is known that, in a Banach space, a sequence (x_n) that is weakly
> convergent (i.e convergent for the weak topology) is not necessarily
> strongly convergent (i.e ||x_n - x|| tends to 0).
> In a Hilbert space, there is the following criterion : if x_n is
> weakly tends to x, and if ||x_n|| tends to ||x|| (as a real sequence),
> then x_n strongly tends to x.
>
> My question is : Is this criterion still true in any Banach space ? If
> not, is there any conter-example ? Is there any condition about the
> space to suppose in order to have this criterion (e.g it has to be
> reflexive...) ?
>

A survey of rotundity and smoothness conditions is in Chapter 7 of
M.M. Day, NORMED LINEAR SPACES. Your convergence property
is called there (H). Some of the results:
locally uniformly rutund (LUR) implies (H).
l^1 can be renormed to be (LUR), so (LUR) does not imply reflexive,
and therefore (H) does not imply reflexive.
Weakly compactly generated (WCG) spaces can be renormed to be (LUR),
in particular all separable spaces and all reflexive spaces can be so
renormed. 

-- 
Gerald A. Edgar              edgar at math.ohio-state.edu
Department of Mathematics    telephone: 614-292-0395 (Office)
The Ohio State University      614-292-4975 (Math. Dept.)
Columbus, OH 43210             614-292-1479 (Dept. Fax)

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