I am trying to understand kernels but have a weak understanding of function spaces. Is there a simple explanation for why the following kernel has the given corresponding functional space and the norm? Is there a method of finding functional spaces and norms for other kernels , say a radial kernel?
Define a linear kernel function as
$k(x,x')= x^Tx'$
on the vector space $X=R^P$
The corresponding functional space is the space of the linear function
$f: R^d -> R: $
$H= \lbrace f(x) = w^Tx : w \in R^p \rbrace $
and the associated norm is just the slope of the linear function
$||f||_H=||w||$