where li this is well-defined as long as > li which we assume from now on let be the linear space of functions of the form > ai xi where > fxi ixi we > ai /li as every function (xcan be represented as see that is linear subspace of (xon define the inner product fxi ihgxi fgih : > li with this inner productthe squared norm of > ai xi is > /li we show that is actually an rkhs with kernel by verifying the conditions of definition firstx kx li xi (xxi hi> as li by assumptionand so is finite secondthe reproducing property holds namelylet > ai xi thenp hk ih hk xi ih fxi > li li xi (xai > li ai xi (xf (xi> the discussion above demonstrates that kernels can be constructed via ( in fact(under mild conditionsany given reproducing kernel can be written in the form ( )where this series representation enjoys desirable convergence properties this result is known as mercer' theoremand is given below we leave the full proof including the precise conditions toe [ ]but the main idea is that reproducing kernel can be thought of as generalization of positive semidefinite matrix kand can also be written in spectral form (see also section in particularby theorem we can write vdvwhere is matrix of orthonormal eigenvectors [vand the diagonal matrix of the (positiveeigenvalues [ ]that isx (ijlv(ivj`> in ( belowxx play the role of ijand xplays the role of vtheorem mercer let be reproducing kernel for compact set then (under mild conditionsthere exists countable sequence of non-negative numbers {ldecreasing to zero and functions {xorthonormal in (xsuch that (xx lx(xx( for all xx ( `> where ( converges absolutely and uniformly on furtherif then (lxis an (eigenvalueeigenfunctionpair for the integral operator (xl (xdefined by [ ]( : (xyf (ydy for