On density and dentability in Hilbert spaces

This paper establishes necessary and sufficient conditions for density and dentability in infinite-dimensional complex Hilbert spaces, demonstrating their fundamental connections to operator theory and optimization through a synthesis of spectral decomposition methods and compact operator approximations. The main result characterizes dentable closed convex subsets $H_0\subset H$ as precisely those whose extreme points form a weakly dense subset, leveraging the Radon–Nikodým Property of Hilbert spaces, and proves the preservation of dentability under countable intersections, finite tensor products, and Cartesian products. The technical framework integrates geometric functional analysis with operator theory, showing that every dentable subset contains a dense separable subspace and that convex dentable operator sets are densely defined. Applications include convergence analysis of gradient-based optimization in function spaces, as well as implications for quantum computing architectures, high-dimensional data analysis, and computational geometry. These theoretical developments also open new directions for spatial modeling and astrophysical density analysis via Hilbert space methods.