A kernel is a function that computes the dot product of two vectors in a new, often higher-dimensional feature space without explicitly transforming the vectors into that space. This is known as the "kernel trick." A kernel function, commonly denoted by $K(\vec{x}_1, \vec{x}_2)$, acts as an implicit feature mapping and provides a measure of similarity or distance between inputs. In the dual formulation of Support Vector Machines (SVMs), replacing the standard dot product with $K(\vec{x}^i, \vec{x}^j)$ allows the model to efficiently learn non-linear decision boundaries in the original feature space. While closely related to the general notion of an inner product, a kernel need not strictly be an inner product.