From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Embedding L2 into dual Banach space
Date: 24 Sep 2001 20:25:17 GMT
Newsgroups: sci.math

In article <3BABAEE5.B7DB013E@noos.fr>, gourio  <gourio@noos.fr> wrote:
>what do you think of the following?
>the dual of  l_1 is isomorphic to l_oo (banach space of bounded
>sequences) (see below); then l_2 is a (proper) subspace of l_oo.

You still have to show that l_2 is (isomorphically) a subspace of 
l_infinity.
The obvious embedding of l_2 into l_infinity is not an isomorphism.

In fact, it turns out that every separable Banach space is isometric
to a subspace of l_infinity.  But the case of l_2 is easy:

Let {x_j} be a sequence that is dense in the unit ball of l_2.  For 
x in l_2, let Tx in l_infinity be defined by (Tx)_k = <x_k, x>
(where <.,.> is the l_2 inner product). 

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2