Hilbert Spaces and Dynamical Systems A Hilbert space is a complete inner product space, often infinite-dimensional (e.g. function spaces). Dynamical systems can be studied in finite-dimensional Euclidean spaces (like 𝑅 𝑛 R n ), but many applications (e.g. PDEs, control theory, optimization, quantum mechanics) naturally live in Hilbert spaces. For example, the Schrödinger equation, gradient flows in function spaces, and wave equations all define dynamical systems in Hilbert spaces.