Reproducing Kernel Hilbert Spaces Cannot Contain all Continuous Functions on a Compact Metric Space

TL;DR

This paper proves that for any uncountable, compact metric space, no reproducing kernel Hilbert space can encompass all continuous functions defined on it, highlighting fundamental limitations in kernel methods.

Contribution

It establishes a theoretical limitation of reproducing kernel Hilbert spaces in representing all continuous functions on uncountable compact metric spaces.

Findings

No RKHS can contain all continuous functions on uncountable compact metric spaces.

Highlights fundamental limitations of kernel methods in function approximation.

Provides a theoretical boundary for the expressiveness of RKHS.

Abstract

Given an uncountable, compact metric space, we show that there exists no reproducing kernel Hilbert space that contains the space of all continuous functions on this compact space.