I understand that the point of using the kernel trick is to project the problem onto a higher dimensional space, where the problem is linearly separable. In this explanation, https://www.quora.com/What-is-the-kernel-trick, it states that the inner product $\langle x,y \rangle$ will be equal to $\langle \phi(x),\phi(y)\rangle$. Understanding this equality seems to be key to understanding how this trick works.
My question is, how do know that our function $\phi$ will preserve the inner product and what are the conditions for this to happen?
I have tried google searching this and despite many references to the kernel trick, I do not believe that this has been answered anywhere.