Editing Mercer's theorem (section)

== Generalizations ==
Mercer's theorem itself is a generalization of the result that any [[symmetric matrix|symmetric]] [[positive-semidefinite matrix]] is the [[Gramian matrix]] of a set of vectors.

The first generalization replaces the interval [''a'',&nbsp;''b''] with any [[compact Hausdorff space]] and Lebesgue measure on [''a'',&nbsp;''b''] is replaced by a finite countably additive measure &mu; on the [[Borel algebra]] of ''X'' whose support is ''X''.  This means that &mu;(''U'') > 0 for any nonempty open subset ''U'' of ''X''.

A recent generalization replaces these conditions by the following: the set ''X'' is a [[first-countable]] topological space endowed with a Borel (complete) measure &mu;. ''X'' is the support of &mu; and, for all ''x'' in ''X'', there is an open set ''U'' containing ''x'' and having finite measure. Then essentially the same result holds:

'''Theorem'''. Suppose ''K'' is a continuous symmetric  positive-definite kernel on ''X''. If the function &kappa; is ''L''<sup>1</sup><sub>&mu;</sub>(''X''), where &kappa;(x)=K(x,x), for all ''x'' in ''X'', then there is an [[orthonormal set]]
{''e''<sub>i</sub>}<sub>i</sub> of ''L''<sup>2</sup><sub>&mu;</sub>(''X'') consisting of eigenfunctions of ''T''<sub>''K''</sub> such that corresponding
sequence of eigenvalues {&lambda;<sub>i</sub>}<sub>i</sub> is nonnegative.  The eigenfunctions corresponding to non-zero eigenvalues are continuous on ''X'' and ''K'' has the representation

:<math> K(s,t) = \sum_{j=1}^\infty \lambda_j \, e_j(s) \, e_j(t) </math>

where the convergence is absolute and uniform on compact subsets of ''X''.

The next generalization deals with representations of ''measurable'' kernels.

Let (''X'', ''M'', &mu;) be a &sigma;-finite measure space.  An ''L''<sup>2</sup> (or square-integrable) kernel on ''X'' is a function

:<math> K \in L^2_{\mu \otimes \mu}(X \times X). </math>

''L''<sup>2</sup> kernels define a bounded operator ''T''<sub>''K''</sub> by the formula

:<math> \langle T_K \varphi, \psi \rangle = \int_{X \times X} K(y,x) \varphi(y) \psi(x) \,d[\mu \otimes \mu](y,x). </math>

''T''<sub>''K''</sub> is a compact operator (actually it is even a [[Hilbert–Schmidt operator]]).  If the kernel ''K'' is symmetric, by the [[compact operator on Hilbert space#Compact self adjoint operator|spectral theorem]], ''T''<sub>''K''</sub>  has an orthonormal basis of eigenvectors. Those eigenvectors that correspond to non-zero eigenvalues can be arranged in a sequence {''e''<sub>''i''</sub>}<sub>''i''</sub> (regardless of separability).

'''Theorem'''.  If ''K'' is a symmetric positive-definite kernel on (''X'', ''M'', &mu;), then

:<math> K(y,x) = \sum_{i \in \mathbb{N}} \lambda_i e_i(y) e_i(x)  </math>

where the convergence in the  ''L''<sup>2</sup> norm.  Note that when continuity of the kernel is not assumed, the expansion no longer converges uniformly.