I am attempting to use SVMs for my class project. For this project, I have selected the gaussian kernel as, well, the kernel. That is,

$$
k(\mathbf{x}_1, \mathbf{x}_n) = e^{-\gamma ||\mathbf{x}_1 - \mathbf{x}_n ||^2}
$$

What I do not understand, is how this kernel is then 'written as a dot-product'. How do we get around doing that? This is because my professor says that when we finalize the training, we will be performing a dot-product between a new vector and the SVs. But given this kernel, how is this dot-product being done?