Editing Mercer's theorem (section)

== Details ==
We now explain in greater detail the structure of the proof of
Mercer's theorem, particularly how it relates to [[spectral theory of compact operators]].

* The map ''K''  ↦ ''T''<sub>''K''</sub> is  [[Injective function|injective]].
* ''T''<sub>''K''</sub> is a non-negative symmetric  compact operator on ''L''<sup>2</sup>[''a'',''b'']; moreover ''K''(''x'', ''x'') &ge; 0.

To show compactness, show that the image of the [[unit ball]] of ''L''<sup>2</sup>[''a'',''b''] under ''T''<sub>''K''</sub> is [[equicontinuous]] and apply [[Ascoli's theorem]], to show that the image of the unit ball is relatively compact in C([''a'',''b'']) with the [[uniform norm]] and ''a fortiori'' in ''L''<sup>2</sup>[''a'',''b''].

Now apply the [[spectral theorem]] for compact operators on Hilbert
spaces to ''T''<sub>''K''</sub> to show the existence of the
orthonormal basis {''e''<sub>i</sub>}<sub>i</sub> of
''L''<sup>2</sup>[''a'',''b'']

:<math> \lambda_i e_i(t)= [T_K e_i](t)  = \int_a^b K(t,s) e_i(s)\, ds. </math>

If &lambda;<sub>i</sub> &ne; 0, the eigenvector ([[eigenfunction]]) ''e''<sub>i</sub> is seen to be continuous on [''a'',''b''].  Now

:<math> \sum_{i=1}^\infty \lambda_i |e_i(t) e_i(s)| \leq \sup_{x \in [a,b]} |K(x,x)|, </math>

which shows that the sequence

:<math> \sum_{i=1}^\infty \lambda_i e_i(t) e_i(s) </math>

converges absolutely and uniformly to a kernel ''K''<sub>0</sub> which is easily seen to define the same operator as the kernel ''K''.  Hence ''K''=''K''<sub>0</sub> from which Mercer's theorem follows.

Finally, to show non-negativity of the eigenvalues one can write <math>\lambda \langle f,f \rangle= \langle f, T_{K}f \rangle</math> and expressing the right hand side as an integral well-approximated by its Riemann sums, which are non-negative
by positive-definiteness of ''K'', implying <math>\lambda \langle f,f \rangle \geq 0</math>, implying <math>\lambda \geq 0 </math>.