From: taga@news.uni-rostock.de (Dr. Michael Ulm) Subject: Re: Does this hold? Date: 10 Dec 2001 13:43:25 +0100 Newsgroups: sci.math Summary: Banach-Alaoglou theorem (compactness of balls in Banach spaces) On 10 Dec 2001 00:58:06 -0800, Felix Goldberg wrote: >Hello, > >I am wondering whether the following assertion is true: > >"Every closed ball in a norm space is sequentially compact under the >metric induced from the norm". > >My hunch is that this is true, and can be shown somehow by >"simulating" R^n on a norm space, but I can't figure out how. > >Any suggestions and hints will be appreciated. > There is a well known result in functional analysis stating that a closed ball in a Banach space is compact if and only if the space is finite dimensional. For a simple example consider the space of summable sequences with the norm given by || x || = sum_i | x_i |. Take as sequence in this space e_n ,where e_n is the sequence wich is 1 at position n and 0 everywhere else. Then || e_n || = 1 for all n, but there does not exist a converging subsequence, since || e_n - e_m || = 2 for n not equal to m. What _is_ true however is the Theorem of Banach-Alaoglou (sp?) which says that if the Banach space in question in a dual space, then each ball is relatively weak-* compact. Any good introductory book on functional analysis should cover that. HTH, Michael. -- &&&&&&&&&&&&&&&&#@#&&&&&&&&&&&&&&&& Dr. Michael Ulm FB Mathematik, Universitaet Rostock taga@hades.math.uni-rostock.de