converselysuppose that evaluation functionals are bounded then from the riesz representation theorem there exists some gdx such that hggdx ig for all -the representer of evaluation if we define (xx gdx ( for all xx xthen : ( *gdx is an element of for every and hgk ig ( )so that the reproducing property in definition is verified the fact that an rkhs has continuous evaluation functionals means that if two functions gh are "closewith respect to kg then their evaluations ( ) (xare close for every formallyconvergence in kg norm implies pointwise convergence for all the following theorem shows that any finite function can serve as reproducing kernel as long as it is finitesymmetricand positive semidefinite the corresponding (unique!rkhs is the completion of the set of all functions of the form pn = ai xi where ai for all theorem moore-aronszajn given non-empty set and any finite symmetric positive semidefinite function rthere exists an rkhs of functions with reproducing kernel moreoverg is unique proof(sketchas the proof of uniqueness is treated in exercise the objective is to prove existence the idea is to construct pre-rkhs from the given function that has the essential structure and then to extend to an rkhs in particulardefine as the set of finite linear combinations of functions xn : ai xi xn xai rn = define on the following inner producth fgig :* = ai xi = : ai (xi = = then is an inner product space in factg has the essential structure we requirenamely that (ievaluation functionals are bounded/continuous (exercise and (iicauchy sequences in that converge pointwise also converge in norm (see exercise we then enlarge to the set of all functions for which there exists cauchy sequence in converging pointwise to and define an inner product on as the limit fgig :lim fn gn ig ( nwhere fn and gn to show that is an rkhs it remains to be shown that ( this inner product is well defined( evaluation functionals remain boundedand ( the space is complete detailed proof is established in exercises and