For fixed $x_1,\ldots,x_n$ and coefficients $c_i$, every $k_m$ satisfies
$$
\sum_{i,j}c_ic_jk_m(x_i,x_j)\geq0.
$$
The sum is finite, so pointwise convergence permits passage to the limit:
$$
\sum_{i,j}c_ic_jk(x_i,x_j)
=\lim_{m\to\infty}\sum_{i,j}c_ic_jk_m(x_i,x_j)
\geq0.
$$
Symmetry also passes to the limit. Hence \b[a pointwise limit of kernels is a kernel].


Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 205 / 3 / 2 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-205/3/2/solution