From: ullrich@math.okstate.edu
Newsgroups: sci.math
Subject: Re: Riesz theorem
Date: Fri, 13 Nov 1998 20:40:50 GMT

In article <364C6E5A.B4D44282@cti.ecp.fr>,
  Daniel Wehsarg <wehsard1@cti.ecp.fr> wrote:
> Is there anybody out there how could please tell me
> a) what the "Riesz theorem" is like and
> b) in how far it asserts that linear functions on a Hilbert space can be
> identified with vectors via the Hilbert space inner product?

           There's a variety of theorems that I've seen called
"the Riesz representation theorem" in various places. If K is a
compact Hausdorff space then the dual of C(K) is the space
of regular Borel complex measures on K; this is what was called
the Riesz representation theorem when I was young. We also
know that the dual of L^p is L^q under certain conditions;
I've seen this fact called the Riesz representation theorem,
although not very often. This last result implies that the
dual of L^2 is L^2; now if you believe that any Hilbert space
has an orthonormal basis it follows that the dual of any
Hilbert space is itself.

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From: israel@math.ubc.ca (Robert Israel)
Newsgroups: sci.math
Subject: Re: Riesz theorem
Date: 13 Nov 1998 22:11:54 GMT

In article <364C6E5A.B4D44282@cti.ecp.fr>, Daniel Wehsarg <wehsard1@cti.ecp.fr> writes:
|> Is there anybody out there how could please tell me
|> a) what the "Riesz theorem" is like and
|> b) in how far it asserts that linear functions on a Hilbert space can be
|> identified with vectors via the Hilbert space inner product?

There are lots of theorems that have the name Riesz attached to them, and
several known as the "Riesz Representation Theorem", but the particular 
one you're asking about in (b) says that every bounded linear functional
on a Hilbert space is given by the inner product with a vector in the
Hilbert space.  Frigyes Riesz was the first to prove this for general Hilbert
spaces (without assuming separability): F. Riesz, "Zur Theorie des Hilbertschen
Raumes", Acta Sci. Math. Szeged 7, 34-38 (1934).  But before general
Hilbert spaces were invented, it was proven for L^2 (which
is not any easier than the general case) by Riesz and M. Frechet 
(independently) in 1907.

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