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 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. -----------== Posted via Deja News, The Discussion Network ==---------- http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own ============================================================================== 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 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