definition reproducing kernel hilbert space for non-empty set xa hilbert space of functions with inner product **ig is called reproducing kernel hilbert space (rkhswith reproducing kernel ifreproducing kernel hilbert space for every xk : ( *is in (xxfor all for every and gg(xhgk ig the reproducing kernel of hilbert space of functionsif it existsis uniquesee exercise the main (thirdcondition in definition is known as the reproducing property this property allows us to evaluate any function at point by taking the inner product of and as suchk is called the representer of evaluation furtherby taking and applying the reproducing propertywe have hk ig ( )and so by symmetry of the inner product it follows that (xx ( xas consequencereproducing kernels are necessarily symmetric functions moreovera reproducing kernel is positive semidefinite functionmeaning that for every and every choice of an and xn xit holds that ai (xi ( reproducing property positive semidefinite = = in other wordsevery gram matrix associated with is positive semidefinite matrixthat is aka for all the proof is addressed in exercise the following theorem gives an alternative characterization of an rkhs the proof uses the riesz representation theorem also note that in the theorem below we could have replaced the word "boundedwith "continuous"as the two are equivalent for linear functionalssee theorem theorem continuous evaluation functionals characterize rkhs an rkhs on set is hilbert space in which every evaluation functional (xis bounded converselya hilbert space of functions for which every evaluation functional is bounded is an rkhs proofnote thatsince evaluation functionals are linear operatorsshowing boundedness is equivalent to showing continuity given an rkhs with reproducing kernel ksuppose that we have sequence gn converging to gthat is kgn gkg we apply the cauchy-schwarz inequality (theorem and the reproducing property of to find that for every and any np | gn |gn (xg( )|hgn gk ig kgn gkg kk kg kgn gkg hk ig kgn gkg (xxnoting that (xxby definition for every xand that kgn gkg as we have shown continuity of that is | gn as for every evaluation functional