This is the identity that I think defines a kernel. $\kappa(x,y) = <\Phi(x),\Phi(y)>$ $\Phi(\cdot)$ is a function that projects the vectors into a feature space. $\kappa(\cdot,\cdot)$ is the associated kernel function.

My understanding of a Kernel is: An inner product between two real vectors both projected into a higher dimensional feature space, which can instead be performed implicitly in a lower dimensional space. The "kernel function" here is a function that performs this implicit calculation. This probably doesn't describe everything that it is to be a Kernel, although I am still interested in that topic, my primary interest is a solid proof, which may indeed involve that topic.