When we kernelize our features, we are implicitly doing a feature mapping of the form $f : \mathbb{R}^n \to \mathbb{R}^m$. These mappings are similar to those used in GANs. When using feature mappings we explicitly compute $f(\vec{x})$ for each $\vec{x}$ in our feature space.

For example if we are given the feature space $\vec{x} = [x_1, x_2]^T \in \mathbb{R}^2$, and we assume that our feature space is not linearly separable, but is quadratically separable, we might define a feature mapping to be $f: \mathbb{R}^2 \to \mathbb{R}^3$ where $f([x_1, x_2]^T) = [x_1^2, \ \ x_2^2, \ \ 1]^T$