 Solution

A positive-semidefinite kernel on $\mathcal{X}$ is a symmetric function $k:{\mathcal{X}}^2\to \mathbb{R}$ such that, for every $x_1,\dots ,x_n$ and $c_1,\dots ,c_n$,If $k(x,x^{\prime })=⟨\varphi (x),\varphi (x^{\prime })⟩$ for a feature map into an inner-product space, thenThus every feature-map inner product is a kernel. If ${\alpha }_1,{\alpha }_2\ge 0$, then each finite quadratic form for ${\alpha }_1{k}_1+{\alpha }_2{k}_2$ is the corresponding nonnegative linear combination, so nonnegative linear combinations of kernels are kernels.