Is there any relationship between a kernel function and the concept of basis in a vector space?
For example does an equality like this
$$ g(x) = \int_{\Omega} \alpha(y) K(x,y)dy $$
have anything to do with something like
$$ g(x) = \sum_{i}\alpha_iK_i(x) $$ ?
Is there any concept of "uncountable" basis maybe?