Confusion related to reproducible kernel hilbert space I am confused why the kernel is called reproducible in kernel hilbert space. Aren't there any easy ones that could help me get some idea what it is all about
 A: In a reproducing kernel Hilbert space $K$, the reproducing kernel $\phi$ "reproduces" functions pointwise by 
$$(f, \phi(\cdot, x)_K = f(x)$$
You can actually build your own reproducing kernel Hilbert space by choosing some subset $\Omega \subset \mathbb{R}^n$, a positive definite function, such as $\phi(x,y) = exp(-\|x-y\|^2)$ and considering the set of all functions $\{k(\cdot,x) : x \in \Omega\}$. You can define a bilinear form on this set by $(k(\cdot,x),k(\cdot,y)) = k(x,y)$, and then take the completion of this space to get a complete inner product space. You can show that the bilinear form is an inner product, and the completion of this pre-Hilbert space is called a Native space for the kernel $k$. See, Holger Wendland's book "Scattered Data Approximation" or Greg Fasshauer's book "Meshfree Approximation Methods in Matlab" for more details.
For a concrete example, consider the Sobolev space $H^1(\mathbb{R})$ with the inner product
$$(f,g)_{H^1} = \int_{\mathbb{R}} f(x) g(x) + f'(x) g'(x) dx$$
Then, the kernel $K(\cdot,x) = \frac{1}{2}exp(-|x-\cdot|)$ is a reproducing kernel, as you can show that
$$(f,k(\cdot,x))_{H^1} = f(x)$$
