Reproducing Kernel Hilbert Spaces

Hilbert Space are Normed Vector Spaces equipped with an inner product. if a kernel is a generalization of an inner product, what is the more general class of spaces that are equipped with kernels which the hilbert space would be a member of

The concept you’re referring to hinges on the generalization of inner product spaces via the introduction of a kernel function. A kernel, in the context of functional analysis and machine learning, is a function that enables the computation of the inner product in an implicitly defined feature space without having to compute the coordinates of the data in that space directly. This concept leads to the framework of Reproducing Kernel Hilbert Spaces (RKHS), which are indeed a more general class of spaces equipped with kernels. However, to clarify, while RKHS are Hilbert spaces associated with a specific kernel function, the kernel concept itself does not necessarily generalize Hilbert spaces but rather extends the utility and application of inner products in these spaces.

Reproducing Kernel Hilbert Spaces (RKHS)

An RKHS is a Hilbert space of functions in which evaluation at each point is a continuous linear functional. Each RKHS is associated with a positive definite kernel function $K(x,y)$ that has the property of reproducing the values of functions in that space:

$$f(x)=⟨f,K(x,\cdot )⟩$$

for all $f$ in the RKHS, where $K(x,\cdot )$ denotes the function in the RKHS associated with the kernel $K$ fixed at $x$. This kernel function essentially generalizes the concept of an inner product by defining it in potentially infinite-dimensional feature spaces.

Kernel Methods and Spaces

In a broader context, when discussing spaces equipped with kernels, one might also consider the realm of kernel methods in machine learning and statistics, where kernels are used to implicitly map data into high-dimensional feature spaces to perform linear algebra operations (such as inner products) in those spaces. These methods rely on the kernel trick, a technique that leverages kernel functions to compute inner products in high-dimensional spaces efficiently, without explicitly performing the mapping.

Relation to Other Spaces

While Hilbert spaces and RKHS are intimately connected through the concept of kernels, the question of more general classes of spaces equipped with kernels touches on several areas:

• Function Spaces with Kernel Functions: Beyond RKHS, various function spaces might be considered in relation to kernels, especially in the context of integral operators and their eigenfunctions. However, RKHS uniquely combines the properties of kernels with the completeness and inner product structure of Hilbert spaces.
• Banach Space: In a different direction, one might also explore Banach spaces, which are complete normed vector spaces not necessarily equipped with an inner product. Certain subspaces of Banach spaces or related constructs might utilize kernels in the analysis of functions or operators, although they lack the direct association with a kernel function that characterizes RKHS.

In summary, RKHS represents a specific and profound integration of the kernel concept with the structure of Hilbert spaces, providing a powerful framework for analyzing and processing functions. While kernels can be used in various mathematical and computational contexts, the unique properties of RKHS highlight the special relationship between kernels and Hilbert spaces, making RKHS a central concept in applications requiring the generalization and extension of inner products.