Kernel Trick and Inner Product Preservation 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.
It doesn't mean that $\langle x, y \rangle = \langle \phi(x), \phi(y)\rangle$.

Kernel method in general means that for an algorithm that involve $\langle x, y\rangle$, we can replace it with a function $K(x,y)=\langle \phi(x), \phi(y)\rangle$ where computing $K(x,y)$ is easy. It is known that for $K$ that satisfies Mercer's theorem, there is a corresponding $\phi$ which is the map to the higher dimensional space.

We do not need to know $\phi$ explicitly and $\phi$ can be very complicated.