Reproducing Kernel Hilbert Spaces and Kernel Metrics

Reproducing Kernel Hilbert Spaces (RKHS) provide a rigorous mathematical framework at the heart of modern functional analysis, statistical learning, and kernel methods. This text-based course guides you through foundational concepts, helping you build intuition and mathematical confidence in working with reproducing kernels and metric spaces. Starting from fundamental vector space definitions, you will explore how kernel functions induce geometry on underlying domains and how operator norm distances characterize complex function spaces. You will gain a structured understanding of how RKHS theory underpins algorithms in data analysis, support vector methods, and kernel embeddings. What you will learn: - Understand the core definitions, axioms, and properties of Hilbert spaces and reproducing kernels - Analyze the reproducing property and evaluate kernel functions on continuous domains - Explore how operator norm distances between kernels induce metrics on the underlying space - Apply kernel tricks and feature map representations used in modern machine learning algorithms - Connect RKHS theory to Gaussian processes, regularized risk minimization, and kernel mean embeddings - Evaluate open research directions and structural properties of functional spaces The course begins with essential mathematical terminology and linear algebra concepts before methodically advancing to metric spaces and practical functional analysis. Designed for learners seeking a clear, accessible introduction to kernel theory, this course requires only basic familiarity with linear algebra and calculus. Begin reading today to strengthen your theoretical toolkit for functional mathematics and machine learning.