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 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 = (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