Need help understanding Riesz representation theorem for Reproducing Kernel Hilbert Spaces I'd like some help understanding any of the following proofs of Riesz representation theorem -- whichever is simpler -- or in fact any proof of the theorem.
Proof 1: http://nfist.pt/~edgarc/wiki/index.php/Riesz_representation_theorem
Proof 2: http://www.mathphysics.com/pde/Rieszapp.html
BTW, I am quite intrigued that the mere existence of a bounded linear functional is sufficient for there to exist a reproducing kernel.
As some of you may already be aware, I am computer scientist trying teaching myself a bit of functional analysis in order to better appreciate the mathematical methods I use. As such simple explanations will be appreciated.
Thanks,
Olumide
 A: This was initially meant to be a comment, but became too long. So, firstly, I'd warmly suggest you to adopt and follow a good book on the subject, rather than using on-line material: which is good as a reference for single proofs, but not often enough orgainzed into a whole theory. The basic material on Hilbert spaces is quite easy and intuitive, but needs several little results, that fit better in a book. You may like Rudin's exposition in Real and Complex analysis (chapt. 4), or Akhiezer & Glazman's Theory of linear operators in Hilbert space (chapt. 1), or Halmos' books Introduction to Hilbert Space and A Hilbert space problem book,... &c: there are of course several very good elementary books on the subject: the best is visiting a library and choose yours. That said, as to the Riesz duality theorem, you may follow this path: 


*

*Two non-zero linear functionals on a vector space have the same kernel if and only if they are scalar multiple of each other. This is easy linear algebra. 

*Then, in order to represent a bounded linear functional $f$ on a Hilbert space via scalar product with a vector $u$, that is $f=\phi_u$ where $\phi_u(x):=(x\cdot u)$, the key point is to find $v\in H$ such that $\ker f = \ker \phi_v$ (so the representing vector $u$ will be some scalar multiple of $v$). But $\ker\phi_v$ is by definition the orthogonal of $v$, so if such a vector $v$ exists, it has to be a generator of the line $(\ker f)^\perp$ because of the (nontrivial) duality relation $V=(V^\perp)^\perp$ for closed linear subspaces.

*From the above you are led to consider the orthogonal decomposition you mentioned, and easily conclude.  
However, orthogonal decompositions and projectors are a topic which is important in its own, and deserve a separate little study (see Zen Harper's comment above). You may start from the concept of metric projection on a convex set C of a Hilbert space H (here the completeness and the uniform convexity of the Hilbert norm ensure existence and uniqueness of the point of C minimal distance from a given point of H). Particularizing the convex to be a linear closed space $V$ you'll see (here is the magic of the Hilbert structure) that the corresponding metric projector on $V$ is a bounded linear operator, and gives you the orthogonal decomposition $H=V\oplus V^{\perp}$, and the above mentioned relation). 
A: 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.
