Hilbert and inner product spaces: Theory, visualization, and applications in machine learning

This study investigates the mathematical structure of Hilbert spaces, defined as complete inner product spaces, and their significance in both theoretical and applied contexts. We begin by exploring their foundational properties, including inner products, orthogonality, and completeness, which extend Euclidean geometric concepts to infinite-dimensional settings. Key mathematical tools, including the Cauchy–Schwarz inequality, triangle inequality, polarization identity, and Apollonius identity, are analyzed to highlight the analytical framework of Hilbert spaces and their relationship to normed spaces and Banach spaces. Then we examine practical applications in quantum mechanics, signal processing, and machine learning, where the inner product structure enables techniques like kernel methods, Support Vector Machines, and Principal Component Analysis. We provide MATLAB-based visualizations are provided, illustrating concepts such as projections and orthonormal expansions in computational contexts. This work integrates rigorous mathematical analysis with practical demonstrations, offering valuable insights for students and researchers in mathematics and data science.