The reproducing property is a central feature of Reproducing Kernel Hilbert Spaces (RKHS). It states that for a Hilbert space of functions $H$ on a set $X$ with a reproducing kernel $K$, every function $f$ in $H$ satisfies

where $K_x$ denotes the function $K(x, \cdot)$, and $\langle \cdot,\cdot \rangle_H$ is the inner product in $H$. This property means that the value of $f$ at any point $x$ can be obtained by taking the inner product of $f$ with the kernel function centered at $x$. The reproducing property gives RKHS its name: the kernel "reproduces" the evaluation of functions via the inner product structure of the space.

The importance of the reproducing property lies in its ability to link function evaluation with the geometry of the Hilbert space. This connection allows you to analyze pointwise behavior of functions using the tools of linear algebra and functional analysis.

A fundamental theorem in the theory of RKHS is the continuity of evaluation functionals. The theorem states:

Theorem (Continuity of Evaluation Functionals):
Let $H$ be a RKHS on $X$ with kernel $K$. For any $x$ in $X$, the evaluation functional $\delta_x: H \to \mathbb{R}$ defined by $\delta_x(f) = f(x)$ is continuous (or equivalently, bounded).

Proof:
The continuity of $\delta_x$ means there exists a constant $C_x$ such that for all $f$ in $H$,

By the reproducing property,

Applying the Cauchy–Schwarz inequality for Hilbert spaces,

Thus,

So, $\delta_x$ is continuous with operator norm at most $\|K_x\|_H = \sqrt{K(x, x)}$. This result is nontrivial because, in a general Hilbert space of functions, pointwise evaluation may not be continuous. In fact, the existence of a reproducing kernel is equivalent to the continuity of all evaluation functionals in the space. In many classical function spaces, evaluation at a point is not a continuous operation unless the space is specifically constructed (as in RKHS) to ensure this property.

To build geometric intuition, consider how the reproducing property encodes pointwise evaluation as an inner product. In a RKHS, the value of a function at a point $x$ is given by the inner product with the kernel function $K_x$. This means that the "direction" in the Hilbert space corresponding to evaluating at $x$ is exactly the kernel function centered at $x$. In other words, the evaluation of any function at $x$ is determined by how much the function aligns with $K_x$ in the Hilbert space sense. This perspective is powerful: it transforms the local, pointwise operation of evaluation into a global, geometric operation — an inner product. This geometric encoding is the foundation for many applications of RKHS in analysis and machine learning, where kernels provide a bridge between data points and the functional geometry of the space.