A Hilbert space is a fundamental concept in mathematics and physics, representing a complete and infinite-dimensional vector space equipped with an inner product. This inner product allows for the measurement of angles and distances, making it possible to define concepts like orthogonality and convergence within the space. Hilbert spaces are essential in various fields, including quantum mechanics, where they provide the framework for describing quantum states. In a Hilbert space, every sequence of vectors that converges has a limit that also belongs to the space, ensuring completeness. This property makes Hilbert spaces particularly useful for solving problems in functional analysis and signal processing, where functions can be treated as vectors.