Friday, 18 September 2009

fa.functional analysis - When is a Banach space a Hilbert space?

In this simple note http://arxiv.org/abs/0907.1813 (to appear in Colloq. Math.), Rossi and I proved a characterization in terms of "inversion of Riesz representation theorem".



Here is the result: let X be a normed space and recall Birkhoff-James ortogonality: xinX is orthogonal to yinX iff for all scalars lambda, one has ||x||leq||x+lambday||.



Let H be a Hilbert space and xrightarrowfx be the Riesz representation. Observe that xinKer(fx)perp, which can be required using Birkhoff-James orthogonality:



Theorem: Let X be a normed (resp. Banach) space and xrightarrowfx be an isometric mapping from X to X such that



1) fx(y)=overlinefy(x)



2) xinKer(fx)perp (in the sense of Birkhoff and James)



Then X is a pre-Hilbert (resp. Hilbert) space and the mapping xrightarrowfx is the Riesz representation.

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