Reproducing Kernel Hilbert Spaces and Kernel Metrics

Reproducing Kernel Hilbert Spaces (RKHS) provide a powerful mathematical framework connecting functional analysis, geometry, and modern machine learning algorithms. This text-based course guides you through the theory of RKHS, beginning with core definitions and progressing to the metric geometries induced by reproducing kernels and their practical significance. You will build an intuitive and rigorous understanding of how kernels map data points into structured function spaces, how operator norms define meaningful distances on underlying domains, and how these geometric concepts underpin modern kernel methods and distribution embeddings. What you'll learn: - Understand the fundamental axioms of Hilbert spaces, inner products, and reproducing kernels - Analyze how reproducing kernels induce metric structures and distances on input spaces - Explore operator norm distances and how they characterize function space properties - Apply kernel methods to foundational machine learning tasks such as ridge regression and support vector machines - Examine modern kernel mean embeddings and maximum mean discrepancy for comparing probability distributions - Practice working with classic kernel families, including Gaussian and polynomial kernels The course begins by establishing foundational terminology and essential linear algebra concepts before systematically examining kernel metrics, operator bounds, and modern algorithmic applications through detailed written explanations and worked examples. This course is designed for beginners in functional analysis and machine learning theory who want a clear, step-by-step introduction without needing prior advanced graduate-level coursework. Start reading today to master the mathematical principles behind kernel spaces and their geometry.