Hilbert spaces

Hilbert spaces are abstract mathematical frameworks that extend the idea of familiar geometric spaces, like the 3D space around us, to possibly infinite dimensions. They consist of vectors (which can represent functions, signals, or states) and allow for concepts like distance, angle, and orthogonality (perpendicularity). These spaces are essential in areas such as quantum mechanics and signal processing, where they help describe complex systems and analyze functions rigorously. Despite their abstract nature, they are a natural generalization of the geometry we're accustomed to, enabling precise mathematical treatment of infinite-dimensional phenomena.