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 $f(x) = ; lt x , x_0 gt$ is an inner product ($x_0$ is the representer of the evaluation function)


• for each evaluation function there exists only one $x_0 in H$


• $parallel f parallel ; = ; parallel x_0 parallel$

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

"For every $g$ in a Hilbert space $mathcal{H}$, $L_gf ; = ; lt g , f gt $ defines a continuous linear functional $L_g$. Conversely, every continuous linear functional $L$ in $mathcal{H}$ has a representation $Lf ; = ; lt g_L , f gt$ for some $g_L in mathcal{H}$, 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.