Hilbert's space

A Hilbert space is a mathematical framework used to analyze infinite-dimensional vector spaces equipped with an inner product, which allows measuring angles and lengths. Think of it as an extension of regular geometry to spaces where each point can represent functions or signals rather than just physical points. This structure enables precise manipulation and understanding of complex systems in fields like quantum mechanics and signal processing. Essentially, it provides a rigorous way to work with infinite sets of data or functions as if they were vectors in a geometric space, fostering advanced analysis and problem-solving in mathematics and physics.