(bshow that every evaluation functional on is continuous at the function that ise kg ( ) ( continuity of at all functions then follows automatically from linearity (cshow that is completethat isevery cauchy sequence fn converges in the norm ||| if and are kernels on and ythen ((xy)( ): (xx (yy and kx ((xy)( : (xx ) (yy are kernels on the cartesian product prove this an rkhs enjoys the following desirable smoothness propertyif (gn is sequence belonging to rkhs on xand kgn gkg then (xlimn gn (xfor all prove thisusing cauchy-schwarz let be an rd -valued random variable that is symmetric about the origin (that isx and (-xidentically distributeddenote by is its distribution and ps(tr are eit eit (dxfor rd is its characteristic function verify that (xx ps( is real-valued positive semidefinite function suppose an rkhs of functions from (with kernel kis invariant under group of transformations xthat isfor all fg and we have (if and (iih tg ig fgig show that ( xt (xx for all xx and given two hilbert spaces and gwe call mapping hilbert space isomorphism if it is (ia linear mapthat isa( bgaaf ba(gfor any fg and ab (iia surjective mapand (iiian isometrythat isfor all fg hit holds that fgih ha fagig let (equipped with the usual euclidean inner productand construct its (continuousdual space gconsisting of all continuous linear functions from to ras follows(afor each define gb via gb (xhbxi bxfor all (bequip with the inner product hgb gg ig :bg show that defined by (bgb for is hilbert space isomorphism let be an model matrix show that xx gi for is invertible as example clearly illustratesthe pdf of random variable that is symmetric about the origin is not in general valid reproducing kernel take two such iid random variables and with common pdf and define denote by psz and fz the characteristic function and pdf of zrespectively show that if psz is in ( )fz is positive semidefinite function use this to show that (xx fz ( {| }( | |/ is valid reproducing kernel hilbert space isomorphism