if $\langle x,y\rangle$ is a Mercer kernel, then is $\langle c_1 x_1 + c_2 x_2,y\rangle$ a Mercer kernel where $c_1+c_2=1$?
Ans: I give the following (dirty) line of proof. Please tell whether its OK or not. $\langle x,y\rangle_K = \langle \Phi(x),\Phi(y)\rangle_H$ in a reproducing kernel Hilbert space. where $x\mapsto\Phi(x)$ is used using a feature map.
Hence, choosing $x_1$ and $x_2$ with probabilities $c_1,c_2$ is equivalent to choosing $\Phi(x_1)$ and $\Phi(x_2)$ with probabilities $c_1,c_2$. As $\Phi(\cdot)$ is one to one, $\langle c_1 x_1 + c_2 x_2,y\rangle$ is a kernel.
Either this is OK or I am doing something really stupid.
