Functional spaces are mathematical constructs used in various fields, particularly in functional analysis. They consist of sets of functions that can be analyzed using the tools of topology and linear algebra. These spaces allow mathematicians to study properties of functions, such as continuity and convergence, in a structured way. One common example of a functional space is the Banach space, which is a complete normed vector space. Another example is the Hilbert space, which is a complete inner product space. Both types of functional spaces are essential in areas like quantum mechanics and signal processing, where they help in understanding complex systems.