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

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- Solution 2

Solution

For fixed

x_{1}, \dots, x_{n}

and coefficients

c_{i}

, every

k_{m}

satisfies

\sum_{i, j} c_{i} c_{j} k_{m} (x_{i}, x_{j}) \geq 0.

The sum is finite, so pointwise convergence permits passage to the limit:

\sum_{i, j} c_{i} c_{j} k (x_{i}, x_{j}) = lim_{m \to \infty} \sum_{i, j} c_{i} c_{j} k_{m} (x_{i}, x_{j}) \geq 0.

Symmetry also passes to the limit. Hence a pointwise limit of kernels is a kernel.

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