The norm in a Reproducing Kernel Hilbert Space (RKHS), often called the RKHS norm, serves as a powerful tool to measure the size and smoothness of functions within the space. Given a function $f$ in an RKHS $H$ associated with a kernel $K$, the RKHS norm is denoted by $||f||_H$. This norm is defined via the inner product in $H$, and it quantifies how "complex" or "rough" a function is in terms of the kernel's structure. Specifically, a smaller RKHS norm indicates that the function is smoother or less oscillatory according to the geometry induced by the kernel. Thus, the RKHS norm not only measures magnitude but also encodes the regularity and smoothness properties that are crucial in applications such as machine learning and signal processing.

The choice of kernel plays a decisive role in determining which functions belong to the RKHS and how smooth they are. For example, consider two widely used kernels: the Gaussian (RBF) kernel and the linear kernel. The Gaussian kernel, defined as $K(x, y) = \exp(-||x - y||^2 / (2\sigma^2))$, induces an RKHS of extremely smooth functions—indeed, functions in this space are infinitely differentiable. In contrast, the linear kernel, given by $K(x, y) = x^T y$, leads to an RKHS consisting of linear functions, which are much less smooth in the sense of higher-order derivatives. The Matérn kernel provides a flexible family where a parameter directly controls the degree of smoothness: higher parameter values produce smoother functions. Thus, by selecting different kernels, you directly influence the regularity and smoothness of the functions in your RKHS, tailoring the space to the requirements of your application.

To gain geometric intuition, imagine the RKHS norm as a way of controlling how "wiggly" or oscillatory a function can be. A function with a small RKHS norm cannot oscillate rapidly; the norm penalizes such behavior according to the kernel's structure. In this sense, the norm acts as a regularizer, suppressing functions that are too irregular or have excessive variation. This property is particularly important in machine learning, where controlling the norm prevents overfitting by favoring smoother, more generalizable solutions. The geometry of the RKHS, as shaped by the kernel and its norm, thus provides a precise mathematical handle on the smoothness of functions.