further reading for good overview of the ridge regression and the lassowe refer the reader to [ for overviews of the theory of rkhs we refer to [ ]and for in-depth background on splines and their connection to rkhss we refer to [ for further details on gp regression we refer to [ and for kernel pca in particular we refer to [ finallymany facts about kernels and their corresponding rkhss can be found in [ exercises let be an rkhs with reproducing kernel show that is positive semidefinite function show that reproducing kernelif it existsis unique let be hilbert space of functions recall that the evaluation functional is the map (xfor given show that evaluation functionals are linear operators let be the pre-rkhs constructed in the proof of theorem thusg is of the form ni= ai xi and hgk ig = ai hk xi ig ai (xi xg(xi= thereforewe may write the evaluation functional of at as :hgk ig show that is bounded on for every xthat is| kg for some continuing exercise let fn be cauchy sequence in such that fn ( ) for all show that fn kg continuing exercises and to show that the inner product ( is well defineda number of facts have to be checked (averify that the limit converges (bverify that the limit is independent of the cauchy sequences used (cverify that the properties of an inner product are satisfied the only non-trivial property to verify is that ff ig if and only if exercises - show that defined in the proof of theorem is an inner product space it remains to prove that is an rkhs this requires us to prove that the inner product space is complete (and thus hilbert)and that its evaluation functionals are bounded and hence continuous (see theorem this is done in number of steps (ashow that is dense in in the sense that every is limit point (with respect to the norm on gof cauchy sequence fn in