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

arXiv: https://arxiv.org/abs/2002.03171 · HTML: https://mlsanity.com/paper/2002.03171/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Feb 2020.

## Built on by (most cited first, 6 of 6)

- [Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks](https://mlsanity.com/paper/2404.18769/) (2024)
- [Topology, Convergence, and Reconstruction of Predictive States](https://mlsanity.com/paper/2109.09203/) (2021)
- [Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators](https://mlsanity.com/paper/2509.26371/) (2025)
- [Shallow ReLUs Networks in Lp-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization](https://mlsanity.com/paper/2605.18468/) (2026)
- [Decomposition of one-layer neural networks via the infinite sum of reproducing kernel Banach spaces](https://mlsanity.com/paper/2409.18132/) (2024)
- [On the Convergence of Irregular Sampling in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2504.13623/) (2025)

## Abstract (arXiv)

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.