To understand how a Reproducing Kernel Hilbert Space (RKHS) is constructed from a given kernel, begin by considering the pre-Hilbert space generated by kernel sections. Given a nonempty set $X$ and a positive definite kernel $K: X × X → ℝ$, you can define, for each $x$ in $X$, a function $K_x: X → ℝ$ by $K_x(y) = K(y, x)$. The collection of all finite linear combinations of these kernel sections forms a vector space:

• Each element is a finite sum of the form $f = ∑_{i=1}^n α_i K_{x_i}$, where $α_i ∈ ℝ$ and $x_i ∈ X$;
• Addition and scalar multiplication are defined pointwise for functions.

To equip this space with an inner product, set

This inner product is well defined because the kernel is positive definite. The resulting space is a pre-Hilbert space: it is a vector space with an inner product, but it may not be complete.

The step-by-step construction proceeds as follows:

• Start with the set of all finite linear combinations of kernel sections;
• Define the inner product as above;
• Verify that this inner product is positive definite and symmetric;
• Observe that the norm induced by the inner product is given by $||f|| = sqrt(⟨f, f⟩)$;
• Recognize that this space may not be complete, so it is only a pre-Hilbert space at this stage.

The next step is to ensure that the function space is complete, which is a requirement for a Hilbert space. This leads to the completion process, which relies on the concept of Cauchy sequences. In this context, a sequence ${f_n}$ in the pre-Hilbert space is Cauchy if, for every $ε > 0$, there exists an $N$ such that for all $m, n ≥ N$, $||f_n - f_m|| < ε$. The space of all such limits, i.e., the completion of the pre-Hilbert space, is the RKHS associated with the kernel $K$.

The following theorem formalizes the existence and uniqueness of the RKHS associated with a positive definite kernel.

Theorem: For every positive definite kernel $K$ on a set $X$, there exists a unique Hilbert space $H$ of functions on $X$ such that:

• For every $x \in X$, the function $K_x(y) = K(y, x)$ belongs to $H$;
• The span of ${K_x: x \in X}$ is dense in $H$;
• The reproducing property holds: for every $f \in H$ and every $x \in X$, $f(x) = \langle f, K_x \rangle_H$.

Proof Outline:

• Start by forming the pre-Hilbert space of finite linear combinations of kernel sections, as described above, with the specified inner product;
• Complete this space with respect to the norm induced by the inner product, yielding a Hilbert space;
• Show that the evaluation functional $f \mapsto f(x)$ is continuous for each $x$, and that the reproducing property holds;
• Uniqueness follows from the fact that any two Hilbert spaces with these properties are isomorphic via an isometry that preserves the kernel functions.

The completion process is crucial for forming the RKHS. The pre-Hilbert space, although equipped with an inner product, might not contain the limits of all Cauchy sequences. By considering all Cauchy sequences and formally adding their limits, you obtain a complete inner product space — a Hilbert space. Every element of the RKHS can be represented as the limit of a sequence of finite linear combinations of kernel sections, and the inner product and norm extend naturally to these limits. This ensures that the RKHS is not only algebraically rich but also topologically complete, allowing you to use powerful Hilbert space techniques.

The geometric intuition behind this construction is that the kernel $K$ encodes the geometry and topology of the function space. The kernel determines how functions are compared, how "close" they are, and how they interact via the inner product. The choice of kernel shapes the space: it controls the smoothness, complexity, and even the types of functions that can be represented in the RKHS. The topology induced by the inner product norm reflects the relationships encoded by the kernel, so that the space is tailored to the properties you wish to model or analyze. In this way, the kernel acts as a blueprint for the geometry of the RKHS.