= Solution
A <positive-semidefinite kernel> on $\mathcal X$ is a symmetric function $k:\mathcal X^2\to\mathbb R$ such that, for every $x_1,\ldots,x_n$ and $c_1,\ldots,c_n$,
$$
\sum_{i,j=1}^nc_ic_jk(x_i,x_j)\geq0.
$$
If $k(x,x')=\langle\phi(x),\phi(x')\rangle$ for a <feature map> into an <inner-product space>, then
$$
\sum_{i,j}c_ic_jk(x_i,x_j)
=\left\|\sum_ic_i\phi(x_i)\right\|^2\geq0.
$$
Thus every feature-map inner product is a kernel. If $\alpha_1,\alpha_2\geq0$, then each finite quadratic form for $\alpha_1k_1+\alpha_2k_2$ is the corresponding nonnegative linear combination, so \b[nonnegative linear combinations of kernels are kernels].
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