I understood that Mercer's Theorem extends the definition of kernels also for infinite input space.

In Machine Learning realm our training set is always finite and hence the input space is always finite. So why are we interested in infinite input spaces?

Where is the flaw in my reasoning?

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Mercer's Theorem and infinite-dimensional spaces aren't used directly. It justifies use of things like the Gaussian kernel in SVMs. Mercer's theorem says this kernel is just an inner product in some other space, but we need not figure out what that space is, or a mapping to it. The fact that it exists is essential to proving that the SVM still works.

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