Functional spaces are mathematical constructs used in various fields, particularly in functional analysis. They consist of sets of functions that share specific properties, allowing mathematicians to study their behavior and relationships. These spaces can be finite-dimensional, like Euclidean spaces, or infinite-dimensional, such as Hilbert spaces and Banach spaces. In functional spaces, functions can be treated as points in a geometric space, enabling the application of linear algebra techniques. This framework is essential for understanding concepts like convergence, continuity, and linear transformations, which are crucial in areas such as quantum mechanics and signal processing.