October  UTC  Ps the delta function I use in the example above as  is the Kronecker delta not the Dirac delta The Kronecker delta is in  whereas the Dirac delta is not in L  Incidentally in response to the request for examples LX is not a RKHS for X any open subset of  On the other hand Hardy spaces usually are RKHSs using various variants of the Szego kernel  For example  the Hilbert space of L functions on the circle is not a RKHS On the other hand  the space of L functions on the circle which are tangential limits of holomorphic functions on the disc is