Monday, 21 August 2006

fa.functional analysis - Need help understanding Riesz representation theorem for Reproducing Kernel Hilbert Spaces

This really isn't an answer. Like Pietro's, it's a comment that got out of hand.



I've been reading a number of books on and offline (thanks to Google books), and I now understand what the kernel of a linear operator is as well as the orthogonal projection theorem, but an understanding of the proof still eludes me. (By the way I've noted that almost all the proofs I've found are versions of each other.) Nevertheless, reading so much about the proof has shed some light on the nature of RKHS, such as:



  • any linear evaluation function f(x)=;ltx,x0gt is an inner product (x0 is the representer of the evaluation function)

  • for each evaluation function there exists only one x0inH

  • parallelfparallel;=;parallelx0parallel

Furthermore, according to "Smoothing Spline ANOVA Models" Gu, Chong 2002 (page 27)



"For every g in a Hilbert space mathcalH, Lgf;=;ltg,fgt defines a continuous linear functional Lg. Conversely, every continuous linear functional L in mathcalH has a representation Lf;=;ltgL,fgt for some gLinmathcalH, called the representer of the evaluation."



This statement demystifies RKHS by the assertion that: every linear evaluation functional is (merely) an inner product of the representer and an element of the RKHS, with the result that the Riesz representation is increasingly seems to to be a definition i.e. something to be accepted and not a result that must be derived.

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