Re: Riesz sequence = isomorphic embedding?

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 06/25/04 

Date: Fri, 25 Jun 2004 07:17:24 -0500

On 25 Jun 2004 01:52:14 GMT, israel@math.ubc.ca (Robert Israel) wrote:

>In article <f5547a6f.0406241727.462352ae@posting.google.com>,
>Alex Gittens <bonobo@myrealbox.com> wrote:
>>I'm stumped by this statement:
>
>>A Riesz sequence in H (a Hilbert space) is the image of the unit
>>vector basis in l_2 (the space of square summable sequences) under an
>>isomorphic embedding.
>
>>What is an isomorphic embedding? (there is no mention of either of
>>those terms up to the point where the statement is made).
>
>It's a linear map, in this case from l_2 into H, which is 1-1 and
>has closed range, but not necessarily onto. Thus it has a continuous
>inverse defined on that range.

And hence, to be specific, a Riesz sequence is a sequence (x_j)
such that

  c sum |a_j|^2 <= ||sum a_j x_j||^2 <= C sum |a_j|^2.

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

************************

David C. Ullrich