Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the smallest scales, such as atoms and subatomic particles. It introduces concepts like wave-particle duality, uncertainty principle, and quantum entanglement, which challenge classical intuitions about the behavior of matter and energy.
Hilbert spaces are complete inner product spaces that generalize the notion of Euclidean spaces to infinite dimensions, providing a rigorous framework for the mathematical formulation of quantum mechanics and other functional analysis applications. They allow for the convergence of sequences of functions and support operations like projection and orthogonal decomposition, essential for solving partial differential equations and studying Fourier transforms.