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 → TK is injective.
- TK is a non-negative symmetric compact operator on L2; moreover K(x, x) ≥ 0.
To show compactness, show that the image of the unit ball of L2 under TK equicontinuous and apply Ascoli's theorem, to show that the image of the unit ball is relatively compact in C with the uniform norm and a fortiori in L2.
Now apply the spectral theorem for compact operators on Hilbert spaces to TK to show the existence of the orthonormal basis {ei}i of L2
If λi ≠ 0, the eigenvector ei is seen to be continuous on . Now
which shows that the sequence
converges absolutely and uniformly to a kernel K0 which is easily seen to define the same operator as the kernel K. Hence K=K0 from which Mercer's theorem follows.
Read more about this topic: Mercer's Theorem
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