From: bruck@pacificnet.net (Ronald Bruck)
Subject: Re: Does anyone have an example like this???
Date: Wed, 31 Mar 1999 15:17:30 -0800
Newsgroups: sci.math
Keywords: Characterization of nonreflexive Banach spaces

In article <7dtu7h$d5s$1@nnrp1.dejanews.com>, ullrich@math.okstate.edu wrote:

> In article <37019A53.4EF6BA80@math.brown.edu>,
>   Munju Kim <mjkim@math.brown.edu> wrote:
> > X : an infinite dimensional Banach space.
> >
> > S : a convex closed subset of X which does not have an element of
> > minimal norm.
> >
> > Please tell me.
> 
>      Homework? Consider l_1, the space of absolutely summable
> sequences. Let e_1, e_2, ... be the standard basis vectors.
> The closed convex hull of the vectors (1 + 1/n)*e_n does it.

Actually, any nonreflexive space has this property.  In fact, this is a
necessary AND sufficient condition for a space to be nonreflexive.

But that's quite a bit harder than the homework problem...  (Which might
NOT be a homework problem.  This approximation problem is so pervasive
it's probably popped in the work of hundreds.)

--Ron Bruck
==============================================================================

From: a.visitor@the.asylum (a.visitor)
Subject: Re: Does anyone have an example like this???
Date: Wed, 31 Mar 1999 20:38:44 +0000
Newsgroups: sci.math

In article <37019A53.4EF6BA80@math.brown.edu>, Munju Kim
<mjkim@math.brown.edu> wrote:

> X : an infinite dimensional Banach space.
> 
> S : a convex closed subset of X which does not have an element of
> minimal norm.
> 
> Please tell me.


Let

    X  = { f:[0,1] \to R  |  f is continuous and f(1)=0  }

and equip X with the uniform norm || ||_{\infty}.

Let

    S  = {  f \in X  |  \int_{0}^{1}tf(t)dt = 1  }


Now prove that

 (1)  X is Banach;

 (2)  S is closed and convex;

 (3)  for all f\in S we have

               ||f||_{\infty} > inf_{g\in S} ||g||_\infty},

      i.e. there is no vector in S closest to 0 (or using your terminology:
      S does not have a "minimal element").